Can you cure anxiety or panic disorder without taking prescribed pills?

Meds or no meds. If you wish you can treat anxiety or panic without medication. Medication will only be necessary if your disorder is too severe to for you to benefit from cognitive-behavioral treatment (cbt). Cbt helps master physical skills like breathing, relaxation; thinking skills like not exaggerating outcomes, not overestimating probabilities; and behavioral skills like gradual exposure to situations.
Possibly. But if you're not successful enough without appropriate medication support, it would be imprudent not to use it. Anxiety can be difficult to beat, so don't fight it "with one hand tied behind your back". Use all the resources available to you to achieve good mental health.