There was no revelation of the mystery at Acts 10 yet, and scripture was not completed.
None of them knew 1 Cor 15:1-4 that people can be saved by just believing in the death burial and resurrection of Christ.
Peter himself did not preach that to Cornelius, instead he said in Acts 10 that one need to "work righteousness" in order to be accepted, which is in line with what he will later write to Israel in 1 Peter 4:12-19.
Acts 10:34 Then Peter opened his mouth, and said, Of a truth I perceive that God is no respecter of persons:
35 But in every nation he that feareth him, and worketh righteousness, is accepted with him.
Without Paul's revelations and the completed scriptures, Israel needed signs to believe that someone was saved, so that was a period of time where God confirmed that gentiles can now be saved, without Israel, without the law, thru them speaking in tongues, which astonished all the Jewish believers present.
If you recall the final verse of Mark
20 And they went forth, and preached every where, the Lord working with them, and confirming the word with signs following. Amen.
The Lord was confirmed the preached word, or the spoken word, with signs. it was necessary since they did not have the completed scriptures yet.
But once scripture is completed and Paul's revelation of the mystery completely written down, we are no longer to look for signs. We believe one is saved the moment he believes in the death burial and resurrection of Christ.
That is why using Acts for salvation doctrine is so problematic, it was a transitional time period between Israel's program and the program of the Body of Christ.
Peter's initial instruction to all groups of people; Jewish, Gentile and Samaritans is clearly reflected in Paul's message and actions recorded in scripture:
Acts 18:6-8
6 And when they opposed themselves, and blasphemed, he shook his raiment, and said unto them, Your blood be upon your own heads; I am clean: from henceforth I will go unto the Gentiles.
7 And he departed thence, and entered into a certain man's house, named Justus, one that worshipped God, whose house joined hard to the synagogue.
8
And Crispus, the chief ruler of the synagogue, believed on the Lord with all his house; and many of the Corinthians hearing believed, and were baptized.
Acts 19:1-10
19 And it came to pass, that, while Apollos was at Corinth, Paul having passed through the upper coasts came to Ephesus: and finding certain disciples,
2 He said unto them, Have ye received the Holy Ghost since ye believed? And they said unto him, We have not so much as heard whether there be any Holy Ghost.
3 And he said unto them, Unto what then were ye baptized? And they said, Unto John's baptism.
4 Then said Paul, John verily baptized with the baptism of repentance, saying unto the people, that they should believe on him which should come after him, that is, on Christ Jesus.
5 When they heard this, they were baptized in the name of the Lord Jesus.
6 And when Paul had laid his hands upon them, the Holy Ghost came on them; and they spake with tongues, and prophesied.
7 And all the men were about twelve.
8 And he went into the synagogue, and spake boldly for the space of three months, disputing and persuading the things concerning the kingdom of God.
9 But when divers were hardened, and believed not, but spake evil of that way before the multitude, he departed from them, and separated the disciples, disputing daily in the school of one Tyrannus.
10 And this continued by the space of two years; so that all they which dwelt in Asia heard the word of the Lord Jesus, both Jews and Greeks.
There is one gospel presented to the Jew first and afterward the Gentile as Paul stated in Romans:
Rom 1:16
For I am not ashamed of the gospel of Christ: for it is the power of God unto salvation to every one that believeth; to the Jew first, and also to the Greek.
Paul also stated there is but one faith:
Eph 4:3-5
3 Endeavouring to keep the unity of the Spirit in the bond of peace.
4 There is one body, and one Spirit, even as ye are called in one hope of your calling;
5 One Lord, one faith, one baptism,