Answering a secondary question first: Acts was not written by Paul. The introduction marks it as volume 2 of the Luke-Acts history, traditionally attributed to Paul's companion Luke. Certainly Paul cannot have been the author, since he is always described in the third person. The narrative style is about him, not by him.
The main question is whether Acts 9.17-19 refers to the baptism of the Holy Spirit. I would say no.
The relevant verses read as follows: (All my verse quotes are from the NIV.)
Then Ananias went to the house and entered it. Placing his hands on Saul, he said, “Brother Saul, the Lord—Jesus, who appeared to you on the road as you were coming here—has sent me so that you may see again and be filled with the Holy Spirit.” Immediately, something like scales fell from Saul’s eyes, and he could see again. He got up and was baptized, and after taking some food, he regained his strength. (Acts 9:17-19)
Technically the word "baptism" here is unqualified, so it could mean either spiritual baptism (God by his Spirit uniting a person with Christ and with the church) or water baptism (what the church does as a sign of that union). But to me the most natural meaning is simple water baptism. The surrounding text has a clear physical sense (even though there are hints of related spiritual meaning). Thus Paul was actually blind, and when the scales fell off he could actually see again (Even though the author may also be inviting us to reflect on Paul's spiritual blindness up to that point). Likewise we are to understand that following his baptism he ate physical food and regained strength after three days of fasting (see verse 9). So it seems best to interpret the baptism as a natural water baptism even if in a wider context we cannot fail to be reminded of the baptism in the Spirit.
It's also worth noting that Paul refers to this event later in Acts when he tells the story of his conversion to a crowd in Jerusalem. He quotes Ananias saying:
And now what are you waiting for? Get up, be baptized and wash your sins away, calling on his name. (Acts 22.16)
The relationship here too is more about water (the image of washing sins away) and less about the filling or receiving of the Spirit. That's not to say that the baptism in the Spirit is denied; rather it's a recognition that we don't have to read this particular aspect of baptism into every single text.
If we consider the general pattern in Acts of people being baptised, we see a similar result. There are some examples where water baptism is clearly described. For example, in the story of Philip and the Ethiopian eunuch the eunuch says,
“Look, here is water. What can stand in the way of my being baptized?” ... And he gave orders to stop the chariot. Then both Philip and the eunuch went down into the water and Philip baptized him. (Acts 8.36-38)
Again, in the story of Cornelius and his household the baptism is in water, because it happens after the gift of the Spirit. The Spirit falls on the Gentiles; they speak in tongues; and so Peter says,
“Surely no one can stand in the way of their being baptized with water. They have received the Holy Spirit just as we have.” So he ordered that they be baptized in the name of Jesus Christ. (Acts 10.47-48)
There are some other cases where the baptism is unqualified like Paul, but to me water baptism appears to be in view. I would include these examples as illustrations:
- Lydia (Acts 16:15)
- The Philippian jailor (Acts 16:33)
- Various people at the Corinthian church (Acts 18:8)
And finally there are no cases in Acts where the baptism is explicitly said to be "in the Spirit" or similar words. There are two cases which at first glance might appear to contradict this, but a closer look reinforces my interpretation above. In Acts 1 we read:
On one occasion, while he was eating with them, he gave them this command: “Do not leave Jerusalem, but wait for the gift my Father promised, which you have heard me speak about. For John baptized with water, but in a few days you will be baptized with the Holy Spirit.” (Acts 1.4-5; see also Acts 11.16 which refers back to this saying of Jesus.)
So this passage clearly shows Luke is aware of the baptism in the Spirit. In fact its importance to him is reflected in this reference to Jesus' teaching at the very beginning of Acts. Luke sums up many days of post resurrection teaching by Jesus in just a few sentences, and this is one of them. Then the rest of Acts is God pouring out his Spirit on the church from Pentecost onwards, in fulfillment of Jesus' prophetic words.
And yet, despite the clear framework of Jesus' own saying, Luke at no point uses that language in describing specific conversion stories. Most obviously, only ten days after the ascension of Jesus the Spirit is poured out in power on a church in exactly the way Jesus had said. Luke tells a story of tongues, of flames, of bystanders miraculously hearing words in their own native languages, and of a sermon so empowered by God that 3,000 became Christians in one day. But none of this is labelled anywhere as "baptised in the Spirit". Surely if there was one point in the whole of Acts where that specific phrase was appropriate, it is here. We may or may not read that element into the story, but it will be our reading, not Luke's.
This is not to say that there is no spiritual component to the baptisms. Of course there is a link, but Luke does not make the link by referring directly to baptism in the Spirit. So we read phrases such as "believe and be baptised". We hear of someone who has been baptised then receiving the gift of the Spirit. Acts 2 is perhaps the clearest picture of this:
When the people heard this, they were cut to the heart and said to Peter and the other apostles, “Brothers, what shall we do?”
Peter replied, “Repent and be baptized, every one of you, in the name of Jesus Christ for the forgiveness of your sins. And you will receive the gift of the Holy Spirit. The promise is for you and your children and for all who are far off—for all whom the Lord our God will call.”
I read this as a practical template for thinking about Christian beginnings. To respond requires a personal component (repentance), a church component (baptism) and a God component (the promise of the Spirit). All three of these elements are woven into the Acts narrative in various ways. Sometimes all three elements are present in a particular case; sometimes only one or two elements are mentioned. But the threefold beginnings is an important theme in Acts generally. And it is thus worth noting that the baptism command is passive. It's not what the believer does to himself. It's what the church does to the believer as a sign of belonging. Luke (who as mentioned earlier knew of Jesus' words about baptism in the Spirit) chooses language that distinguishes the church's baptism in water and God's gift of the Holy Spirit. They are closely related in time and in meaning. But for Luke they are not the same, and so when we read Acts neither should we read them as being the same.