Although Cornelius was commander of a hundred soldiers he was still humble and showed great respect for Peter. When Peter entered the house he found a crowd - Cornelius' whole household was gathered to hear the Word of God.
Acts 10:28-29
And he said unto them, Ye know how that it is an unlawful thing for a man that is a Jew to keep company, or come unto one of another nation; but God hath shewed me that I should not call any man common or unclean.
Therefore came I unto you without gainsaying, as soon as I was sent for: I ask therefore for what intent ye have sent for me?
Peter recognized that the vision he had seen on the rooftop in Joppa was preparation for his visit to Cornelius. God had shown him that it was OK for him to visit the Gentiles. At this point however, he had still not quite realized that the Gentiles were about to receive the same gift of holy spirit that the Judean believers had received and manifested on the day of Pentecost. Therefore, Peter asked Cornelius why he had sent for him.
Acts 10:30-33
And Cornelius said, Four days ago I was fasting until this hour; and at the ninth hour I prayed in my house, and, behold, a man stood before me in bright clothing,
And said, Cornelius, thy prayer is heard, and thine alms are had in remembrance in the sight of God.
Send therefore to Joppa, and call hither Simon, whose surname is Peter; he is lodged in the house of one Simon a tanner by the sea side: who, when he cometh, shall speak unto thee.
Immediately therefore I sent to thee; and thou hast well done that thou art come. Now therefore are we all here present before God, to hear all things that are commanded thee of God.
Cornelius recounted the visit from the angel and told Peter that they were all gathered together to hear the Word of God. They were not interested in Peter's opinion - they wanted to know what God had to say. In Acts Chapter 11 when Peter recounts this conversation the record adds that Cornelius told him that God had said that Peter "shall tell thee words, whereby thou and all thy house shall be saved."
Finally, Peter came to the full realization of what was about to happen. The Gentiles wanted to hear the Word of God and God had verified that it was HIS will that they do! Now all Peter has to do is teach Cornelius and his household the same Word that he had been teaching among the Judeans for years: "Jesus Christ is Lord and God raised him from the dead." If it was really available for Gentiles to get saved that is what they needed to hear and believe.
Peter FINALLY realized that God was no respecter of persons and that all who wanted to believe were accepted in this new grace administration. This was a great day for Peter as well as for the Gentiles. So, Peter taught them that Jesus Christ is Lord and that God raised him from the dead. The Gentiles were so hungry to believe and get born again that they acted immediately upon the Word as they heard. This was quite a shock to Peter and the six Judean witnesses he had brought with him.
The Judean believers were astonished! Not by some "confession" they heard from the Gentiles. Not by some feeling they got. They were astonished when they saw and heard the evidence of holy spirit within. The Gentiles SPOKE IN TONGUES! You cannot use something you do not have. The only way a person can manifest holy spirit and speak in tongues is if that person has spirit born within them. The Gentiles heard, believed, and manifested the SAME gift that the Judean believers had received many years earlier on the day of Pentecost. That is what amazed the Judean witnesses. They saw and heard the proof!
Peter and the Judean believers were so rattled by witnessing the new birth in this Gentile household that Peter proposed a water baptism ceremony. Now, the Judeans had continued to keep the law of Moses and the practice of outward ordinances even after being saved by grace on the day of Pentecost. Water baptism is a ceremonial cleansing practiced by John the Baptist prior to the sacrifice and resurrection of the Lord Jesus Christ. John baptized with water, but these Gentiles had already been baptized with holy spirit just as had the Judean believers on the day of Pentecost. Our cleansing is an spiritual internal cleansing with holy spirit. The water baptism ceremony is not required as is evident in this record where the Gentiles had already received salvation and manifested the holy spirit within them. Although Peter started to introduce this legalistic outward ordinance, there is evidence in the next chapter of Acts that he probably did not follow through and water baptize the Gentiles of Cornelius' household.
Acts 10:46-48
For they heard them speak with tongues, and magnify God. Then answered Peter,
Can any man forbid water, that these should not be baptized, which have received the Holy Ghost as well as we?
And he commanded them to be baptized in the name of the Lord. Then prayed they him to tarry certain days.
The key is the Gentiles had already received the SAME gift of holy spirit that the Judeans had received. Baptism with holy spirit - getting born again - cleansed on the inside - that is the true baptism. Now, if you want to dip in water after that I guess that is up to you. But God gave them the gift by grace without any works. We are baptized in the name of Jesus Christ when we confess with our mouth the Lord Jesus and believe in our heart that God raised him from the dead. We are cleansed by receiving holy spirit within. Washing on the outside is nothing but ceremony. Doesn't hurt you, but it sure is not a requirement for salvation.
The Gentiles had received holy spirit and been born again. Peter and the six Judean brethren had witnessed the birth. Now, they had to convince the Apostles, elders, and Judean believers in Jerusalem that the Gentiles had received.
Gladly
Acts