Why did Pete apologize to Tony Stark?

Peter lost his father at a very young age. At this stage, Tony is a father figure to him. More so because Peter sees much in common with Tony, both being men of science, superheroes and children at heart (agreed Peter is quite literally one). When he starts to fade, his first reaction is panic! That's when Peter starts to say "I don't wanna go" over and over. This part is pretty obvious. He is a young kid, way over his head in this battle and now he's dying.

Eventually, he realises that Tony is as helpless as him. He can't do anything to stop this. He can see the desolation and despair in Tony's eyes. It is at this moment that Peter eventually accepts his fate. In his dying moments, he apologises to Tony for failing him. This is a sincere sorry that comes from a son who feels he let his father down.

Tony didn't even want Peter here in the first place. He deployed the spider suit's emergency protocol to get Peter off the spaceship. It was Peter who decided to stay, to help Tony. And now he feels remorse for not listening to him. For he knows, how hard his death will hit Tony. His parting words were both meant as an apology for not doing more as well as for not keeping himself safe.

We can't know exactly, but there are valid (in-character) reasons for Peter to want to apologize to Tony.

1

When they were pulling Thanos' glove off, Peter actually succeeded (it slipped off) but Thanos managed to grab it just in time.

This is a possible source for Peter's apology. If only he had been a bit faster in pulling the glove off, Thanos would not have been able to commit his atrocity.

We, the viewer, know that if anyone were to blame, it's Star Lord for losing his cool. However, keep in mind that Peter (Parker) may not have been observing what happened as he was focused on pulling the glove off; or he could just as well regret not having been faster inspite of Star Lord's mistake. He was so close to pulling it off, after all.

2

Secondly, Peter knows that he went against Tony's wishes by tagging along on the spaceship. Notice Peter and Tony's interaction when Tony finds out he's still on the ship.

Tony You should not be here.
Peter I was gonna go home.
Tony I don't wanna hear it.
Peter But it was such a long way down and I just thought about you on the way...
Tony And now I gotta hear it.
Peter ...and kinda stuck to the side of the ship. And this suit is ridiculously intuitive, by the way.
Tony God damn it.
Peter So, if anything, it's kinda your fault that I'm here.
Tony What did you just say?
Peter I take that back. And now, I'm here in space.
Tony Yeah, right where I didn't want you to be.

When you look at the scene, you can see that Peter regrets saying that it's Tony's fault before Tony responds.

This proves that Peter cares about what Tony thinks, and doesn't want to make Tony feel even more responsible than he already does. And as much as Peter wanted to help, he also acknowledges that Tony isn't happy with Peter's decision because Tony is looking out for him.

This means that he could be apologizing to Tony because he regrets staying on the ship. Not because he's dying, but because he respects Tony so much that he regrets not having listened to him as a matter of principle.

Because of this line that Iron Man had in Spider-Man: Homecoming:

IRON MAN: If you die, I feel like that’s on me.