The simple answer is yes, definitely. He assumed this because all the apostles and other inspired teachers in the early church believed this. Paul did not discuss baptism all that much in his letters because the question of baptism was not an issue in the young church. There was no one in the church questioning the connection between baptism and salvation. Paul’s letters discuss issues with which the church was struggling. Baptism was not one of these. Even so–even though Paul does not spend a lot of time on baptism–we can clearly see his doctrine on baptism from the small number of references to baptism. For example, in 1 Corinthians 12:13 he says "For we were all baptized by one Spirit into one body." Paul is talking about the church here. He reminded the Corinthains that when they were baptized, they were not only saved, but they became part of the body of Christ–the Church. Paul clearly associates baptism with salvation in Galatians 3:27. "all of you who were baptized into Christ have been clothed with Christ." The converse of this is that those who have not been baptized into Christ have not been clothed with Christ. In other words, they are not Christians. In Romans 6:2-7 Paul makes a similar comment. Again, he tells us that we are baptized into Christ. This is about as clear as possible a statement that one must be baptized to be a Christian.
We can debate the definition of "Christian," because it is never exactly defined in the New Testament. However, for me personally, I define a Christian as one who is saved, one who is "in Christ," one who has been forgiven of their sins, and one who has the indwelling gift of the Holy Spirit. If this is the definition of a Christian, then the Bible teaches, and Paul certainly agrees, that all Christians have been baptized.