That then is not a doctrinal difference, but a personal failing on Peter's part. Peter feared those of the 'circumcision' sect (2:12) The doctrine was taken care of in 2:9 as well as in Acts, with Peter being the one who first broke the barrier of evangelization to the Gentiles. Paul in his rebuke notes that Peter didn't have problems UNTIL the circumcism sect showed up, then Peter withdrew.
But looking deeper were the chosen twelve immediately after The Resurrection doing what they were actually told in The Great Commission? Not really. They were staying in Jerusalem living and dwelling with each other until the death of Stephen which began the persecution in earnest and which resulted in The Gospel reaching far places.
Yet it was your claim that the deep divison in DOCTRINE occured in this time frame. How so - Paul was not an apostle until AFTER the death of Stephen.
But your analysis is still flawed. for instance, Peter was in Joppa and went to Caesarea in the break through with the Gentiles (Acts 11:2) thus the assertion that all they did was sit around in Jerusalem is not entirely correct.
pay closer attention to the Great Commission. While the ultimate goal was to reach all nations, Jesus gave further instruction in Acts 1:8 ". . . and ye shall be witnesses unto me both in Jerusalem, and in all Judaea, and in Samaria, and unto the uttermost part of the earth. " Thus Jesus outlined the sequential growth of evangelism. Further, in Acts 2, the world came to Jerusalem and the converts took it back with them to their homes outside of Israel. Phillip converting the Ethopian eunich - with the message carried there as a result. The Holy Spirit guided and directed, but the persecution was not a correction by the same - unless you can prove specifically from the bible that that was specifically stated as the case.
As believers fled the persecution they took the message of The Gospel with them. This was not that immediate as in a matter of days, weeks, etc, but rather between Stephens death and the fall of Rome. Tradition has it before the chosen 12 had died it may have reached as far as Spain.