What did Paul mean when he told the jailer to believe on the Lord Jesus Christ?
Paul meant that believing on the Lord Jesus Christ implies the Gospel: Christ died for your sins and rose again. He did not mention repentance or baptism, because this was a change from the Kingdom program under the Law to Gentile ground under grace.
Les explains that when Paul said 'believe on the Lord Jesus Christ,' he was not contradicting the earlier message but indicating a change of program. At the heart of Paul's message was the Gospel: Christ died for your sins and rose from the dead. So 'believe' implies trusting that He died for you and rose again. Unlike Acts 2, where the question was 'what must we do?' and the answer was 'repent and be baptized,' Paul says nothing about repentance and baptism. This is because the setting is now Gentile ground under grace, whereas the Kingdom program was still under the Law and temple worship had not been stopped.
And they said, Believe on the Lord Jesus Christ, and thou shalt be saved, and thy house.
More Questions
What did Paul tell the Gentile to do to be saved in Acts 16?
When the Gentile in Acts 16 asks 'what must I do to be saved?', Paul answers 'believe on the Lord Jesus Christ and thou shalt be saved'—not 'repent and be baptized.'