We can see from the passage that the only repenting they did was change their mind about the Christ. IOW, the only thing they did to be born again was believe the gospel, and then they were given the spiritual gift of tongues.
Believe and repent... not just believe all by itself.
Repent is turning to God, because you turn away from Satan, and from his kingdom.
This is the way we receive the forgiveness of sin and are born again... not just believe all by itself.
Your doctrine has repent lumped together as meaning the same thing as believe.
Why would the scripture say Repent and believe, two distinctly separate things, which are clearly taught by Paul.
9 that if you confess with your mouth the Lord Jesus
and believe in your heart that God has raised Him from the dead, you will be saved.
Confessing Jesus as Lord, is how we show that we have turned to Him, and away from Satan as out former lord.
The scriptures teach us that is what the house of Cornelius did and received eternal life as a result.
The plain and clear scriptures teach this as plain as day... if you confess with your mouth the Lord Jesus
and believe in your heart that God has raised Him from the dead, you will be saved.
...with the mouth confession is made unto salvation.
Not believing is made unto salvation,
but confession is made unto salvation.
44 While Peter was still speaking these words, the Holy Spirit fell upon all those who heard the word.
There is nothing here to suggest that these did not repent,
even before Peter arrived, as Cornelius was aware of the Gospel message that was being preached throughout the area... as the scriptures said -
And they said, "Cornelius the centurion, a just man, one who fears God and has a good reputation among all the nation of the Jews, was divinely instructed by a holy angel to summon you to his house, and to hear words from you." Obedience to do what the angel instructed, in order that salvation would come to his household...
But in every nation whoever fears Him and works righteousness is accepted by Him. The word which God sent to the children of Israel, preaching peace through Jesus Christ--He is Lord of all--
that word you know, which was proclaimed throughout all Judea, and began from Galilee after the baptism which John preached:
Acts 10:35-37
Cornelius just needed the missing ingredient from Paul's Gospel message... And
how shall they hear without a preacher?
And how shall they preach
unless they are sent? Romans 10:14-15
45 And those of the circumcision who believed were astonished, as many as came with Peter, because the gift of the Holy Spirit had been poured out on the Gentiles also.
46 For they heard them speak with tongues and magnify God. Then Peter answered,
47 "Can anyone forbid water, that these should not be baptized who have received the Holy Spirit just as we have?"
Acts 10:44-47
Just like the disciples of John that had already repented, but needed to hear about Jesus...
1 And it happened, while Apollos was at Corinth, that Paul, having passed through the upper regions, came to Ephesus. And finding some disciples
2 he said to them, "Did you receive the Holy Spirit when you believed?" So they said to him, "We have not so much as heard whether there is a Holy Spirit."
3 And he said to them, "Into what then were you baptized?" So they said, "Into John's baptism."
4 Then Paul said, "John indeed baptized with a baptism of repentance, saying to the people that they should believe on Him who would 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 hands on them, the Holy Spirit came upon them, and they spoke with tongues and prophesied.
Acts 19:1-6
We know that the house of Cornelius did repent and receive life...not just believe all by itself.
18 When they heard these things they became silent; and they glorified God, saying, "Then God has also granted to the Gentiles repentance to life."