If Jesus had been sent to the Gentiles, why are they upset if all of them could preach to the Gentiles in Acts 11. Why didn't Peter just mention the Great Commission? (I think that was his objection).
The Pharisees that Peter was going to did not need to know the exact words of the Great Commission.
Peter doesn't mention The Great Commission because he is being asked a specific question.
"11 Now the apostles and the brothers[a] who were throughout Judea heard that the Gentiles also had received the word of God. 2 So when Peter went up to Jerusalem, the circumcision party[b] criticized him, saying, 3 “You went to uncircumcised men and ate with them.” 4 But Peter began and explained it to them in order: 5 “I was in the city of Joppa praying, and in a trance I saw a vision, something like a great sheet descending, being let down from heaven by its four corners, and it came down to me. 6 Looking at it closely, I observed animals and beasts of prey and reptiles and birds of the air. 7 And I heard a voice saying to me, ‘Rise, Peter; kill and eat.’ 8 But I said, ‘By no means, Lord; for nothing common or unclean has ever entered my mouth.’ 9 But the voice answered a second time from heaven, ‘What God has made clean, do not call common.’ 10 This happened three times, and all was drawn up again into heaven. 11 And behold, at that very moment three men arrived at the house in which we were, sent to me from Caesarea. 12 And the Spirit told me to go with them, making no distinction. These six brothers also accompanied me, and we entered the man's house. 13 And he told us how he had seen the angel stand in his house and say, ‘Send to Joppa and bring Simon who is called Peter; 14 he will declare to you a message by which you will be saved, you and all your household.’ 15 As I began to speak, the Holy Spirit fell on them just as on us at the beginning. 16 And I remembered the word of the Lord, how he said, ‘John baptized with water, but you will be baptized with the Holy Spirit.’ 17 If then God gave the same gift to them as he gave to us when we believed in the Lord Jesus Christ, who was I that I could stand in God's way?” 18 When they heard these things they fell silent. And they glorified God, saying, “Then to the Gentiles also God has granted repentance that leads to life.”"
The problem was not going to the Gentiles, the actual problem was that the circumcised party were upset that the Gentiles had not been circumcised and received everything that with it as necessary to becoming Christians (Information that Anthony Rogers brought to my attention when I posed the question on my personal Facebook).
To add to this, The Gentiles through faith in the Messiah now have equal access to God without having to become Jews and that they are not lesser than Jews.
Is the phrase by my mouth limiting it by Peter's mouth only and that no one else can speak it?
No. Let's read:
"Acts 15:6 The apostles and the elders were gathered together to consider this matter. 7 And after there had been much debate, Peter stood up and said to them, “Brothers, you know that in the early days God made a choice among you, that by my mouth the Gentiles should hear the word of the gospel and believe. 8 And God, who knows the heart, bore witness to them, by giving them the Holy Spirit just as he did to us, 9 and he made no distinction between us and them, having cleansed their hearts by faith. 10 Now, therefore, why are you putting God to the test by placing a yoke on the neck of the disciples that neither our fathers nor we have been able to bear? 11 But we believe that we will be saved through the grace of the Lord Jesus, just as they will.”"
The phrase "By my mouth" isn't limiting Peter to being the only one who can proclaim to the Gentiles and absolutely no one else, that is a gross abuse of the text and is refuted by the following later in Luke's letter:
"12 And all the assembly fell silent, and they listened to Barnabas and Paul as they related what signs and wonders God had done through them among the Gentiles. 13 After they finished speaking, James replied, “Brothers, listen to me. 14 Simeon has related how God first visited the Gentiles, to take from them a people for his name. 15 And with this the words of the prophets agree, just as it is written,
16 “‘After this I will return,
and I will rebuild the tent of David that has fallen;
I will rebuild its ruins,
and I will restore it,
17 that the remnant[b] of mankind may seek the Lord,
and all the Gentiles who are called by my name,
says the Lord, who makes these things 18 known from of old.’
19 Therefore my judgment is that we should not trouble those of the Gentiles who turn to God, 20 but should write to them to abstain from the things polluted by idols, and from sexual immorality, and from what has been strangled, and from blood. 21 For from ancient generations Moses has had in every city those who proclaim him, for he is read every Sabbath in the synagogues.”"
Not only were signs and wonders were done by Barnabas and Paul but the Gentiles would have also listened to them and by no means does Peter raise any objections and say "By my mouth Paul the Gentiles here the message of the Gospel, not yours". The Acts 15 Council are deciding which laws Gentiles are bound to keep as more of them were being admitted into the growing church. Peter cannot be possibly saying that he alone is the one to speak to the Gentiles, All the disciples had that right in light of the Great Commission. Jesus said to his disciples ""
Paul claims he was sent to the Gentiles and Peter was to the Jews and Peter says he was to go to the Gentiles. Was there a conflict between Peter and Paul on who went to go to the Gentiles?
Peter in his own letters is speaking to Jewish believers whereas Paul spoke to mixed to Gentile congregations but as was his habit, he would go to the synagogue first and then to the Gentiles afterwards, although some of the synagogues would contain Gentile hearers as well, namely ones similar to Cornelius who believed in the God of Israel but had not converted to Judaism yet. Acts 17 shows us this particular custom of Paul's evangelism and life: https://www.biblegateway.com/passage/?search=Acts+17
There was also a case in Acts 18 where Paul washed his hands of a group of Jews because of the way he was treated:
"18 After this, Paul left Athens and went to Corinth. 2 There he met a Jew named Aquila, a native of Pontus, who had recently come from Italy with his wife Priscilla, because Claudius had ordered all Jews to leave Rome. Paul went to see them, 3 and because he was a tentmaker as they were, he stayed and worked with them. 4 Every Sabbath he reasoned in the synagogue, trying to persuade Jews and Greeks.
5 When Silas and Timothy came from Macedonia, Paul devoted himself exclusively to preaching, testifying to the Jews that Jesus was the Messiah. 6 But when they opposed Paul and became abusive, he shook out his clothes in protest and said to them, “Your blood be on your own heads! I am innocent of it. From now on I will go to the Gentiles.”
7 Then Paul left the synagogue and went next door to the house of Titius Justus, a worshiper of God. 8 Crispus, the synagogue leader, and his entire household believed in the Lord; and many of the Corinthians who heard Paul believed and were baptized."
This is consistent with what Jesus said about wiping the dust of your feet when they refuse to receive the Gospel:
"Matthew 10:1 Jesus called his twelve disciples to him and gave them authority to drive out impure spirits and to heal every disease and sickness.
2 These are the names of the twelve apostles: first, Simon (who is called Peter) and his brother Andrew; James son of Zebedee, and his brother John; 3 Philip and Bartholomew; Thomas and Matthew the tax collector; James son of Alphaeus, and Thaddaeus; 4 Simon the Zealot and Judas Iscariot, who betrayed him.
5 These twelve Jesus sent out with the following instructions: “Do not go among the Gentiles or enter any town of the Samaritans. 6 Go rather to the lost sheep of Israel. 7 As you go, proclaim this message: ‘The kingdom of heaven has come near.’ 8 Heal the sick, raise the dead, cleanse those who have leprosy,[a] drive out demons. Freely you have received; freely give.
9 “Do not get any gold or silver or copper to take with you in your belts— 10 no bag for the journey or extra shirt or sandals or a staff, for the worker is worth his keep. 11 Whatever town or village you enter, search there for some worthy person and stay at their house until you leave. 12 As you enter the home, give it your greeting. 13 If the home is deserving, let your peace rest on it; if it is not, let your peace return to you. 14 If anyone will not welcome you or listen to your words, leave that home or town and shake the dust off your feet. 15 Truly I tell you, it will be more bearable for Sodom and Gomorrah on the day of judgment than for that town."
While this was before the Great Commission, where Jesus would send his disciples out among the Gentiles, there is an important point here. If someone consistently refuses you or will not listen to the words of the Gospel, you leave, wipe the dust of your feet and go elsewhere to someone who will listen.
Back to the letters of the New Testament, we see who the writers are addressing at the very start of their letters.
Here are the two openings to Peter's letters:
"1 Peter, an apostle of Jesus Christ,
To those who are elect exiles of the Dispersion in Pontus, Galatia, Cappadocia, Asia, and Bithynia, 2 according to the foreknowledge of God the Father, in the sanctification of the Spirit, for obedience to Jesus Christ and for sprinkling with his blood:
May grace and peace be multiplied to you."
"1 Simeon[a] Peter, a servant[b] and apostle of Jesus Christ,
To those who have obtained a faith of equal standing with ours by the righteousness of our God and Savior Jesus Christ:
2 May grace and peace be multiplied to you in the knowledge of God and of Jesus our Lord."
See James' first line in his letter
"1 James, a servant[a] of God and of the Lord Jesus Christ,
To the twelve tribes in the Dispersion:
John in his letters, 2 and 3 John addresses different individuals too.
2 John: "1 The elder to the elect lady and her children, whom I love in truth, and not only I, but also all who know the truth, 2 because of the truth that abides in us and will be with us forever:
3 Grace, mercy, and peace will be with us, from God the Father and from Jesus Christ the Father's Son, in truth and love."
3 John "1 The elder to the beloved Gaius, whom I love in truth.
2 Beloved, I pray that all may go well with you and that you may be in good health, as it goes well with your soul. 3 For I rejoiced greatly when the brothers[a] came and testified to your truth, as indeed you are walking in the truth. 4 I have no greater joy than to hear that my children are walking in the truth."
Paul in his letters would either address an individual (Titus, 1 and 2 Timothy and Philemon) or address various churches (Galatians, Ephesians, Phillipians, Colossians, 1 and 2 Thessalonians, 1 and 2 Corinthians, Romans and some debate Hebrews).
It is also worth noting in Galatians 2 that at one stage, Paul was sent to the Gentiles, whereas Peter was to go the Jews.
"2 Then after fourteen years I went up again to Jerusalem with Barnabas, taking Titus along with me. 2 I went up because of a revelation and set before them (though privately before those who seemed influential) the gospel that I proclaim among the Gentiles, in order to make sure I was not running or had not run in vain. 3 But even Titus, who was with me, was not forced to be circumcised, though he was a Greek. 4 Yet because of false brothers secretly brought in—who slipped in to spy out our freedom that we have in Christ Jesus, so that they might bring us into slavery— 5 to them we did not yield in submission even for a moment, so that the truth of the gospel might be preserved for you. 6 And from those who seemed to be influential (what they were makes no difference to me; God shows no partiality)—those, I say, who seemed influential added nothing to me. 7 On the contrary, when they saw that I had been entrusted with the gospel to the uncircumcised, just as Peter had been entrusted with the gospel to the circumcised 8 (for he who worked through Peter for his apostolic ministry to the circumcised worked also through me for mine to the Gentiles), 9 and when James and Cephas and John, who seemed to be pillars, perceived the grace that was given to me, they gave the right hand of fellowship to Barnabas and me, that we should go to the Gentiles and they to the circumcised. 10 Only, they asked us to remember the poor, the very thing I was eager to do."
The context doesn't mean they couldn't address the opposite groups as we observe that Paul often went to the Jews first then to the Gentiles. But here we have a period of time where Paul would speak to mostly Gentiles, leaving the Jews to the Apostles. Does this mean that Paul, Peter, James and John never addressed others outside of a specific group? No, they can and could address anyone if the Lord led them to do it.
More maybe added later if the Lord Wills.