In Acts 10, the conversion of Cornelius, Peter begins with,
(Acts 10:34-35) Then Peter began to speak: “I now realize how true it is that God does not show favoritism 35 but accepts men from every nation who fear him and do what is right.”
This is a missional statement, indeed the very basis for mission activity. The language goes all the way back to God’s covenant with Abraham and his promise to bless all nations through Abraham’s seed.
(Acts 10:36) “You know the message God sent to the people of Israel, telling the good news of peace through Jesus Christ, who is Lord of all.”
Now, “good news” is a term that predates the ministry of Jesus, and Peter is plainly using it in its Old Testament sense. It refers to the ascension of a king, but it also refers to the Old Testament prophecies that speak in terms of serving the poor, such as —
(Isa 61:1-2) The Spirit of the Sovereign LORD is on me, because the LORD has anointed me to preach good news to the poor. He has sent me to bind up the brokenhearted, to proclaim freedom for the captives and release from darkness for the prisoners, 2 to proclaim the year of the Lord’s favor and the day of vengeance of our God, to comfort all who mourn,
which is a passage Jesus applied to himself. According to Isaiah (and Jesus), “good news” is good news for the weak and vulnerable of society.
More precisely, Peter’s point is that the prophesied kingdom has dawned, a kingdom in which the poor and needy would be cared for and sins would be forgiven and God would grant peace to his people by saving them from their enemies.
(Acts 10:37-38) “You know what has happened throughout Judea, beginning in Galilee after the baptism that John preached — 38 how God anointed Jesus of Nazareth with the Holy Spirit and power, and how he went around doing good and healing all who were under the power of the devil, because God was with him.”
Peter next explains that Jesus “went around doing good and healing.” Again, this is a root of our mission. Peter doesn’t begin with “You are sinner and Jesus saves.” Rather, he talks about what Jesus did in serving others — which is entirely consistent with the Jewish understanding of “good news” — as well as Cornelius’ own life.
In fact, we should backtrack just a bit to notice why God chose Cornelius to be the first Gentile convert —
(Acts 10:1-2) At Caesarea there was a man named Cornelius, a centurion in what was known as the Italian Regiment. 2 He and all his family were devout and God-fearing; he gave generously to those in need and prayed to God regularly.
Manifestly, Cornelius was a Roman soldier. To the Jews, he was part of an occupying force. But Luke describes him as God-fearing, indeed, as one who gave generously to those in need.
Why is this important? It’s important in that Jews saw concern for the poor as a critically important part of their religion. The Law of Moses includes scores of commands about caring for the poor. The prophets repeatedly command concern for the poor. For Cornelius to be seen as God-fearing, he had to be concerned for the poor.
In verses 39-41, Peter preaches the resurrection of Jesus. And then, in 42, he teaches that Jesus will judge us all.
(Acts 10:43) “All the prophets testify about him that everyone who believes in him receives forgiveness of sins through his name.”
It’s not until verse 43 that Peter gets to forgiveness of sins. And even then, he speaks in terms of forgiveness as spoken of by the prophets. This is because forgiveness of sins is a sign of the Kingdom, God’s reign, and the fulfillment of the many promises found in the Old Testament prophets. To get a true flavor of Peter’s teaching, we have to see what the prophets actually said on the subject —
(Jer 31:33-34) “This is the covenant I will make with the house of Israel after that time,” declares the LORD. “I will put my law in their minds and write it on their hearts. I will be their God, and they will be my people. 34 No longer will a man teach his neighbor, or a man his brother, saying, ‘Know the LORD,’ because they will all know me, from the least of them to the greatest,” declares the LORD. “For I will forgive their wickedness and will remember their sins no more.”
This passage is among the earliest “forgiveness” passages in the prophets. It’s clearly speaking of the Messianic age and is quoted in Hebrews 8, where the author begins an extensive commentary on it.
But notice this — this is not personal forgiveness. It’s forgiveness of a community — the house of Israel, that is, the church as the church. The call to Cornelius is to become part of the forgiven community, the new Israel.
See, similarly,
(Jer 33:7-8) I will bring Judah and Israel back from captivity and will rebuild them as they were before. 8 I will cleanse them from all the sin they have committed against me and will forgive all their sins of rebellion against me.
(Jer 50:19-20) But I will bring Israel back to his own pasture and he will graze on Carmel and Bashan; his appetite will be satisfied on the hills of Ephraim and Gilead. 20 In those days, at that time,” declares the LORD, “search will be made for Israel’s guilt, but there will be none, and for the sins of Judah, but none will be found, for I will forgive the remnant I spare.
(Micah 7:18) Who is a God like you, who pardons sin and forgives the transgression of the remnant of his inheritance? You do not stay angry forever but delight to show mercy.
In each case, the promised forgiveness is all about God’s relationship with the elect community, not individuals. Of course, the individuals are saved, but it’s because they become part of the people God has forgiven.
Thus, while Peter most certainly teaches forgiveness of sins, he teaches it as a part of the fulfillment of God’s mission promised through the prophets and realized in Jesus, not only to forgive sins, but in “doing good and healing all who were under the power of the devil.” Peter is calling Cornelius to become a part of the forgiven community through which God will realize the Kingdom prophecies.
Now, does Peter tell Cornelius he is to go preach Jesus and serve the needy? Not in so many words. But he preaches the coming of the prophesied Kingdom in these very terms. And talks about God’s mission to spread the word about Jesus to all nations long before he gets to forgiveness of sins.
In other words, Peter’s gospel sermon is about much, much more than personal salvation. Indeed, it’s about the salvation of God’s elect community, about becoming a part of that community, and about good news, promised centuries before, in which God will create a nation — a community — where the poor and mourning will be cared for.
It’s missional.
Let's be more precise. This statement was not given as incentive for Cornelius to respond to the gospel. The statement was that Peter now realized that he should be bringing the gospel to the Gentiles. That is, it was Peter who was being persuaded by this fact, not Cornelius. Peter was not offering Cornelius a mission. He was explaining why Cornelius was eligible to receive the gospel.
Next, Peter demonstrated that Jesus was the Messiah, showing how he fulfilled prophecy. (verses 36-43 etc). Cornelius knew the prophecies, the "message to the people of Israel." Peter was showing him that Jesus was the fulfilment of the prophetic message. He offered eyewitness testimony to the resurrection, and told of the judgment and of forgiveness through Jesus.
There is not one word in Peter's message calling on Cornelius to carry out a mission. (As you pointed out yourself, Cornelius was already doing good to the poor, prior to being taught the gospel.) Cornelius was compelled to respond by the truth of the message (proven by prophecy, the resurrection, and eyewitness testimony), by the prospect of judgment, and by the offer of forgiveness…. not by some supposed invitation to being part of a mission.
Jesus did teach his disciples to give to the poor. I think that can even be called part of our mission. But that mission was never offered as a reason why a non-Christian should convert and follow Jesus — not in Cornelius' case, nor any other case in scripture.