Can my gynecomastia go away on its own?

Yes. Most cases resolve on their own over a period of several months, especially if it is associated with prescription medications that can be substituted.
Sometimes. Most cases of pubertal gynecomastia resolves spontaneously, without the need for intervention. Gynecomastia in older patients however usually does not resolve spontaneously. I have found that gynecomastia, if persistent, is often best treated with partial excision of the prominent glandular tissue as well as liposuction surgery of the peripheral chest area.
No. Most gynecomastia cases involve breast tissue which does not go away on it's own.