Will a herniated disk go away by itself?

Possibly. If the herniation is acute (just happened) a common course is that these issues resolve on their own within 6-8 weeks. Often physical therapy and anti-inflammatories are important in the healing process. If the pain does not resolve with these conservative measures or is accompanied by lower extremity weakness/numbness you should seek the care of an expert as soon as possible.
Most of the time-yes. Most disc herniations resolve in terms of pain --90% without surgery. When seen on a MRI and that is repeated at six months, 60% of these disc herniations have resorbed or are smaller than as seen on the initial mri.