Tuggy's argument fails for a number of reasons. For one thing, Peter's sermon in Acts 2 also makes no mention of Christ's death being an atoning sacrifice for our sins. Extending Tuggy's logic further, therefore, would require also abandoning that doctrine as definitional to the gospel.
The second problem is that, while Peter's sermon (being addressed to a Jewish audience) does not use the philosophical categories that would be developed later to convey the idea of the Trinity, Peter's sermon is thoroughly Trinitarian. Consider verses 16-21, in which Peter quotes from Joel 2:
But this is what was uttered through the prophet Joel: “‘And in the last days it shall be, God declares, that I will pour out my Spirit on all flesh, and your sons and your daughters shall prophesy, and your young men shall see visions, and your old men shall dream dreams; even on my male servants and female servants in those days I will pour out my Spirit, and they shall prophesy. And I will show wonders in the heavens above and signs on the earth below, blood, and fire, and vapor of smoke; the sun shall be turned to darkness and the moon to blood, before the day of the Lord comes, the great and magnificent day. And it shall come to pass that everyone who calls upon the name of the Lord shall be saved.’Notice that in verses 17 and 18, Yahweh states that "I will pour out my Spirit". Yet what does Peter go on to state in verses 32-33?
This Jesus God raised up, and of that we all are witnesses. Being therefore exalted at the right hand of God, and having received from the Father the promise of the Holy Spirit, he has poured out this that you yourselves are seeing and hearing.Thus, the one who has poured out the Spirit, according to Peter, is Jesus Himself! Peter thus has identified Jesus as none other than Yahweh. Jesus, moreover, is clearly distinct from the Father, since Peter says that He has "received from the Father."
Furthermore, in Acts 2:21, Peter quotes Joel as saying,
And it shall come to pass that everyone who calls upon the name of the Lord shall be saved.And yet what does Peter state in verse 38?
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.Thus, while Peter has quoted Joel as saying that all who call upon the name of Yahweh will be saved, he goes on to instruct the people to repent and be baptized in the name of Jesus Christ.
Indeed, Peter goes on to say to the Jewish council in Acts 4:11-12,
This Jesus is the stone that was rejected by you, the builders, which has become the cornerstone. And there is salvation in no one else, for there is no other name under heaven given among men by which we must be saved.Again, the name given by which we must be saved is that of Jesus.
We thus have clear evidence in this text that Peter is proclaiming the deity of Jesus. But what of the Holy Spirit? In Acts 2:17, Peter has quoted God in the book of Joel as saying "I will pour out my Spirit on all flesh." What properties would the Holy Spirit have to possess in order to simultaneously be poured out on all flesh all over the world? Clearly, omnipresence, an attribute of deity.
The book of Acts actually affirms the deity and personal identity of the Holy Spirit a number of times. For example, in Acts 5:3-4, we read,
But Peter said, "Ananias, why has Satan filled your heart to lie to the Holy Spirit and to keep back for yourself part of the proceeds of the land? While it remained unsold, did it not remain your own? And after it was sold, was it not at your disposal? Why is it that you have contrived this deed in your heart? You have not lied to man but to God.”Thus, the Holy Spirit is here taken to be a personal agent who can be lied to -- in fact, the Holy Spirit is identified as God Himself in verse 4.
Acts 5:32 also has another affirmation of the personal identity of the Holy Spirit:
And we are witnesses to these things, and so is the Holy Spirit, whom God has given to those who obey him.”Thus, the Holy Spirit is a witness in the same sense that the apostles were witnesses. This again strongly implies the personal identity of the Holy Spirit.
The Holy Spirit is identified as both divine and personal in Acts 13:2:
While they were worshiping the Lord and fasting, the Holy Spirit said, “Set apart for me Barnabas and Saul for the work to which I have called them.”Here, the Holy Spirit speaks as a person and sends Barnabas and Paul. The Holy Spirit says it is the work He has called them to do. This again implies strongly the deity of the Holy Spirit.
The personal identity of the Holy Spirit also comes through in Acts 15:28 at the Jerusalem council:
For it has seemed good to the Holy Spirit and to us to lay on you no greater burden than these requirements.The phrase "it has seemed good to the Holy Spirit" (which is taken to be in the same sense that it seemed good to the apostles) implies strongly that the Holy Spirit is a personal agent.
Another affirmation of the deity of the Holy Spirit occurs in Acts 28:25-27:
And disagreeing among themselves, they departed after Paul had made one statement: “The Holy Spirit was right in saying to your fathers through Isaiah the prophet: “‘Go to this people, and say, “You will indeed hear but never understand, and you will indeed see but never perceive.” For this people's heart has grown dull, and with their ears they can barely hear, and their eyes they have closed; lest they should see with their eyes and hear with their ears and understand with their heart and turn, and I would heal them.’Thus, according to Paul, the one who spoke to Isaiah in Isaiah 6:9-10 was the Holy Spirit. Yet according to Isaiah 6:8 it is the Lord God himself speaking.
In conclusion, we have seen that, contrary to Dale Tuggy's assertions, Peter's sermon in Acts 2 is profoundly Trinitarian.