Head tilt is a very serious illness and shouldn't just be left without treatment to see how it progresses, and very rarely will it spontaneously go away without the correct treatment, especially if it is due to a bacterial ear infection. It is always a symptom of a serious illness. It can even be fatal. I myself have lost a rabbit to head tilt and another one to the e. cuniculi parasite, which can cause head tilt. If not treated correctly, most times it will progressively get worse, til your rabbits head twists so much that your rabbit will start to uncontrollably roll.
If your rabbit does have an ear infection causing this, not giving antibiotics will mean that it will continue to get worse and can even spread. Yes in some cases antibiotics can upset a rabbits gut, especially if the wrong antibiotic is given to a rabbit orally. Usually this is why baytril is the first antibiotic a vet will try with rabbits. Because it is usually well tolerated and causes the least amount of digestive upset. But if this is an ear infection, antibiotics are absolutely needed to have any chance of clearing it up. If the vet wants to verify an inner ear infection, sometimes a head xray will show this.
If your vet isn't confident about knowing how to diagnose head tilt correctly, or when to prescribe the correct antibiotics to rabbits when warranted, you need to find a better vet and one that actually knows what they are doing where rabbits are concerned. Cat and dog vets don't know enough about rabbits. Rabbits are completely different and need a vet that specializes in them, which would be an exotics vet. Personally, if I went to a vet that didn't take this very seriously and treat accordingly, I would be looking for a new vet and would never take my rabbits to that previous vet again.
If you aren't sure what antibiotics are safe for rabbits, these lists show which ones are and aren't. Certain antibiotics are very dangerous to give to rabbits orally, so it is essential to see a vet that knows what they are doing and is an experienced rabbit exotics vet.
http://www.medirabbit.com/Unsafe_medication/dangerous_antibiotics.htm
http://www.medirabbit.com/Safe_medication/Antibiotics/Safe_antibiotics.htm
Read the links I shared. Google 'heat tilt in rabbits'. You will see just how serious this can be. It is one of the dreaded and worst rabbit illnesses there is. It can be extremely difficult to treat and the longer it is left, the harder it can become to stop irreversible cell damage from occurring, or even to save the rabbit. This illness requires prompt treatment, and the sooner it is treated, the better chance you will have of saving your rabbit and preventing permanent damage from occurring. Panacur, antibiotic(baytril), and metacam are the standard treatment for
rabbit knowledgeable exotics vets to give when they can't pinpoint the cause of the head tilt.