In Acts 10, Peter has a vision:

Act 10:11 and [Peter] saw heaven opened and an object like a great sheet bound at the four corners, descending to him and let down to the earth.
Act 10:12 In it were all kinds of four-footed animals of the earth, wild beasts, creeping things, and birds of the air.
Act 10:13 And a voice came to him, “Rise, Peter; kill and eat.”
Act 10:14 But Peter said, “Not so, Lord! For I have never eaten anything common or unclean.”
Act 10:15 And a voice spoke to him again the second time, “What God has cleansed you must not call common.”
Act 10:16 This was done three times. And the object was taken up into heaven again.

Peter wonders what this means. It is important to note that Peter was still following the symbolic law and keeping kosher dietary laws (as evident in this text). This was well after the events in Acts 2, meaning Acts 2 was not a “salvation by faith alone” message. When Peter finally meets the gentile Cornelius, he interprets his vision, showing his shock at Gentiles being accepted:

Act 10:28 Then he said to them, “You know how unlawful it is for a Jewish man to keep company with or go to one of another nation. But God has shown me that I should not call any man common or unclean.

Peter, at this point of time has not even considered that the Gentiles were part of Christ’s promise. It is important to note that Jesus himself focused his ministry on the Jews, excluding Gentiles and half-Jews (Samaritans). It is at this event that Peter starts to believe that Gentiles can be saved through works of the law:

…Act 10:34 Then Peter opened his mouth and said: “In truth I perceive that God shows no partiality.
Act 10:35 But in every nation whoever fears Him and works righteousness is accepted by Him.

Peter, in Acts 9, has been in contact with Paul only 15 days (see Gal 1:18). Paul seems not to have influenced him at this point of time, which is to be expected from a fledgling Christian who the church still feared. That Peter was not influenced (or perhaps the subject was not broached) is made increasingly evident by what comes next:

Act 10:36 The word which God sent to the children of Israel, preaching peace through Jesus Christ—He is Lord of all—
Act 10:37 that word you know, which was proclaimed throughout all Judea, and began from Galilee after the baptism which John preached:

Did John preach “salvation by faith alone”? Did Christ preach “salvation by faith alone”? Peter tells the gentile Cornelius not only to “work righteousness” but to heed the “word which God sent to the children of Israel”. Peter is still preaching the Gospel of the Kingdom, and for good reason; this gospel was reinforced to him the 40 days after Christ rose from the dead while Christ was with the apostles (Act 1:3)). Accepting the Gentiles as equals (along with salvation by faith alone) was a new concept to Peter. Peter next clarifies the gospel:

Act 10:38 how God anointed Jesus of Nazareth with the Holy Spirit and with power, who went about doing good and healing all who were oppressed by the devil, for God was with Him.
Act 10:39 And we are witnesses of all things which He did both in the land of the Jews and in Jerusalem, whom they killed by hanging on a tree.
Act 10:40 Him God raised up on the third day, and showed Him openly,
Act 10:41 not to all the people, but to witnesses chosen before by God, even to us who ate and drank with Him after He arose from the dead.
Act 10:42 And He commanded us to preach to the people, and to testify that it is He who was ordained by God to be Judge of the living and the dead.
Act 10:43 To Him all the prophets witness that, through His name, whoever believes in Him will receive remission of sins.”

Does believe on him entail believing Christ’s “Gospel of the Kingdom” or is it what modern Christians interpret “believing in him” to be? The preceding text, mentioning “righteousness” and the Gospel of “John the Baptist” and “Jesus of Nazareth” indicate the former. Peter’s words, in which he accepts Gentiles, is shocking to everyone who hears:

Act 10:45 And those of the circumcision who believed were astonished, as many as came with Peter, because the gift of the Holy Spirit had been poured out on the Gentiles also.

Throughout this entire incident, Peter does not seem to understand more than the fact that “the Gentiles are not unclean”. He does not advocate “salvation by faith alone” or even current Jews forsaking the symbolic law. This can best be illustrated by the events later described in Galatians 1 and Acts 15.

Paul, though, uses this event as leverage for his own message (Eph 3:5).