No, the whole story of Cornelius and Acts 10 was to SHOW Peter and the other Jews that God had a plan for the long abhorred Gentiles. He showed him in no uncertain terms that the Holy Spirit fell upon the gentile Cornelius just as it did the Jewish disciples at Pentecost. Peter's vision on the roof was not about foods necessarily, but to demonstrate to Peter God's intention with the Gentiles that Jews had drilled into their heads that they were unclean.
Acts 10 is the beginning of the Gentile inclusion. Afterwards, some of the Jews tried to force circumcision on the Gentile converts, so yes Acts is all about the body of Christ. A lot of Acts addresses the natural branches (the Jews), but after Acts 10, it is about the Body of Christ as a whole.
By the way, God does the work of water baptism, by HIS grace through faith in Christ. Without God's work, baptism is just getting wet. Those two things are not opposed as you seem to indicate. It's not water baptism vs grace through faith. It is water baptism by grace through faith in Christ. Baptism cannot and was not parsed separately from faith, Especially in the first century. These days a lot of folks DO try that.