Profanation is the act of saying or doing something terribly offensive or blasphemous. Swearing in a mosque would definitely be considered profanation by your deeply religious family.
When someone violates anything that's considered to be sacred, that's profanation — it's no wonder that you'll most often find this word in a religious context. Disrespecting someone's deeply held religious beliefs is one kind of profanation. Littering at a holy site is another. The words profanation, profanity, and profane all come from the Latin root profanus, "unholy."