The full verse in context is as follows:
26 “therefore I declare to you this day that I am not responsible for the blood of any of you, 27 for I did not shrink from declaring to you the whole purpose of God. 28 Keep watch over yourselves and over all the flock, of which the Holy Spirit has made you overseers, to shepherd the church of God[a]that he obtained with the blood of his own Son.[e] (NRSV Acts 20:26-28)
This verse gives rise to two questions. (1) does this mean Jesus is God? (2) if not, then what does it mean saying that Christ’s blood is actually God’s “own”?
The first question is the easier to deal with. This is one of the 10 or so verses which are claimed for the title “God” being applied to Jesus, and is still used to argue that today. But it is pretty clear from “purpose of God” in 27 and “commend you to God” in 32 that “church of God” also refers to God, not Jesus.
The second is a more serious problem. Is it “bought with the blood of his own” meaning “own Son” (CEV)? Or is it “bought with his own blood” (most versions)?
The reason that most versions go with the the second – “through his own blood” (dia tou idiou haimatos), is that the word “Son” would be required to make that statement clear. And, as unpleasant at might be to the ear, the Greek text does say “bought with his own blood. And yet we know, clearly that it was not God’s own blood, but the blood of his son.
Conclusion
The views of commentators are mixed. Some find this a difficult or provocative statement from Paul, some think it natural. Either way, it is what it is. And if it is meant to emphasize something it is the cost God has paid for the flock he, God, ultimately purchased. If that is so, then best to take the text as it stands.