Are Antibiotics Always Necessary To Get Over An Ear Infection?

No, quite often they aren't.

Most minor infections -- including ear infections -- will sort themselves out, without too much discomfort to the afflicted patient, given enough time.  

Until recently it was thought that most ear infections were best treated with antiBs.  This would prevent (it was hoped) perforation of the ear drum, glue-ear or otherwise undesirable outcomes.  However, the ear drum bursting isn't such a bad thing as long as further infection is prevented.  Moreover, a study in 2004 showed that antibiotics only shortened the term of an ear infection by about two days on average.  Part of the reason is that many ear infections are viral in origin, not bacterial.  Antibiotics are useless against viral infections.

Moreover, it is now recognised that antibiotics have been prescribed too eagerly in the past. which has led to problems with antibiotic resistant bacteria.

There is still some controversary about this, and health professionals are far from united in their approach.  However, it is very common now to hear that an earache or minor ear infection should be treated with analgesics (pain relief) as the first line of defense.  It is only if the infection is progressing badly, taking too long (more than a week) to get better, the sufferer is very young, has a persistant history of ear infections or analgesics are ineffective that antibiotics are now typically prescribed.