Do antibiotics like azithromycin permanently cure acne?

No. They do not permanently cure acne, rather they just control it while you take the medication.
No. There is only one drug that may cure acne-isotretinoin. All the others will control it while you are on it.
No. Nothing cures acne until it self-cures as you get older. Isotretinoin may give you a remission lasting several years.

Related Questions

What's the next best antibiotic class for treating adult acne after doxycycline and bactrim (sulfamethoxazole and trimethoprim)? Is cephalexin or azithromycin a good option?

Use the big guns. An antibiotic should never be used as sole treatment for acne. If you've been doing this, you need to add a topical agent such a benzoyl peroxide. If the combination has failed, it's time for you to consider isotretinoin. It's not for the faint-hearted, but it's better than loading antibiotics that may leave you with colitis or worse. Read more...
Acne treatment. Acne is treated by several different medications. Antibiotics aim to treat common skin bacteria that may precipitate acne formation, but antibiotics are not always the cure. Drying agents, washes, benzoil peroxides may be needed. Some severe cases may require dermatology evaluations and tx such as accutane. Best to see the skin dr. (derm) and get tx. Before using many antibx. Best wishes. Read more...