Why doesn't Adrian Toomes give up Spider-Man's identity?

Because family means more to him and Spider-Man saved his daughter, something he said he will never forget.

Overhearing, Herman Schultz showed Toomes a news report on Spider-Man saving Toomes' daughter.

Source: Vulture > Biography > Vulture's Campaign > Hijacking Damage Control Truck (4th Paragraph)

Toomes: Peter, nothing is more important than family. You saved my daughter’s life. I could never forget something like that. So I’m gonna give you one chance. Are you ready? You walk through those doors, you forget any of this happened. And don’t you ever, ever interfere with my business again. Because if you do, I’ll kill you and everybody you love. I’ll kill you dead. That’s what I’ll do to protect my family. Do you understand?

Source: Spider-Man: Homecoming on Transcripts Wikia, Quote in Vulture > Personality

this is also noted at the end of the bio section on the wikia

As Toomes listened Gargan noted that he had friends on the outside looking to kill Spider-Man and noted that he had heard a rumor that Toomes knew who Spider-Man was. Believing that he still owed Peter Parker, Toomes merely responded to Gargan that if he knew who Spider-Man was, he would already be dead before then leaving to talk with his family.

Adrian owes Peter for saving his daughter Liz and respects Peter for this as shown when he gives Peter the warning to not interfere with his operations when he was delivering Peter and Liz for their date and even after Peter continues to interfere he tried to recruit him to avoid having to kill him.

Adrian Toomes is not a straight-up evil villain bent on world domination or revenge. He becomes a criminal as a means to support his family. His beef with Spider-Man is not personal; he doesn't try to kill Spider-Man because he hates him, but because Spider-Man got in the way of him running his business.

At the end, Toomes is in jail. His plans have failed; he is no longer running a criminal business. Having Spider-Man killed simply wouldn't help him in any way; it wouldn't allow him to keep running his criminal business. He has no motivation to try to have Spider-Man killed.

This is in addition to the fact that he knows Spider-Man and his daughter are friends (and somewhat more), and that Spider-Man saved his daughter's (and his) life.

I'm going to offer an alternate answer based on the facts the movie presents.

It's not because Peter saved his daughter:

Toomes: Peter, nothing is more important than family. You saved my daughter’s life. I could never forget something like that. So I’m gonna give you one chance. Are you ready? You walk through those doors, you forget any of this happened. And don’t you ever, ever interfere with my business again. Because if you do, I’ll kill you and everybody you love. I’ll kill you dead. That’s what I’ll do to protect my family. Do you understand?

At this point in the movie, Peter already saved his daughter's life. Because of this Toomes, offers him a chance to walk away.

However, Peter does not do this, and what immediately happens? Toomes' henchman is in the parking lot waiting for Peter, and arguably would have killed him had Ned not happened to be there. Then Peter goes to the warehouse where he confronts Toomes. There, Toomes tries to kill Peter by knocking the supports of the warehouse roof out with his vulture suit. He nearly succeeds, with Peter trapped and nearly crushed by rubble, and only able to emerge after he confronts his fears about his fate without his suit.

During the final fight it seems clear the Toomes seems conflicted about whether he actually wants to kill Peter, as he goes after his goal to "not come home empty handed" when he could have delivered the death blow while he had Peter down. Clearly, this is not something that you would want everyone to know. It's not his respect for Peter or his wish to not have him dead that causes him to play dumb regarding Peter's identity. It's not about saving Peter, it's about his own family. Peter is the only person he has confided in about his commitment to his family. Giving up Peter's identity compromises him. It places his family at risk, but worse still, it does so while losing face while in prison. How can you have a silent tough guy image - something he had always maintained even with his henchman - when not only were you humiliated by Spiderman, but it turns out Spiderman is a teen aged high schooler who took your own daughter to homecoming and you knew he did and allowed it to happen? Reveal Peter's identity and you risk all that coming out. You win the battle and lose the war.