He's worried himself sick about his daughter.
In this sentence, Is either He's [He is] or [He has] ?
It is He has. This sentence is an example of the present perfect tense. The present perfect is formed with to have (here, He has) and a past participle (here, worried).
You're probably asking "Why can't it be is?" because "He is worried" is also a perfectly good sentence. The answer is that if it were is, then "worried himself sick" doesn't make sense, so it must be has.
The answer can only be "He has". "He is worried himself sick about his daughter" is ungrammatical and makes no sense. To worry oneself sick means to become ill through worry.