Does Leontes become jealous?  Yes.  Does it "destroy" him?  All things considered, since he is reunited with both his wife and daughter at the end of the play, he is not truly destroyed.

By the end of Act III, it would appear that he is destroyed.  In Act III, scene 2, Pauline confronts the jealous tyrant and enumerates all that he has done in his unfounded jealous anger.

In his jealousy, he betrayed his best friend, Polixenes.  She tells him that to send away his baby daughter to be left alone in the wild was an evil action.  She also lays the death of the young prince at Leontes's feet.  Finally she tells him that he is responsible for the death of Hermione.

In rejecting the Oracle's answer, Leontes brings down the wrath of the gods and until what is lost is found (Perdita) Leontes will remain a broken man.

For what it is worth, when Patrick Stewart played the role, he asked a psychiatrist friend to help him understand Leontes who appears to become jealous out of the blues, so to speak.  What his friend discovered was that the jealousy we see is actual a real mental illness that mimics jealousy but progresses in stages, just like Leontes.

First this unfounded jealousy seems to come on suddenly and usually the wife is suspected to have had an affair with the best friend, although there is no evidence.

Secondly there is a desire and attempt to kill the friend and a rejection of the wife.

Thirdly there is a deep depression that can last days, weeks, and even years.

According to Stewart, this information helped him as an actor to understand his character.

Shakespeare was a great observer of his fellow man and this is just another example.