That is actually a more complicated question than you would think.
The human body can get rid of many infections. Back in history, before antibiotics were invented, people usually did not die from minor skin infections. Certainly, a small percentage of minor infections would get progressively worse and could cause a systemic infection. And at that time in history, infectious diseases were a major cause of death. But for minor skin infections, good cleaning and the body's immune system will be able to handle the majority of infections. The advantage of antibiotics is that it will eliminate even the small percentage that will progress to a more serious infection.
Of course, there are serious infections in which antibiotics should be started quickly, such as pneumonia or kidney infections. But for minor infections, it is not as necessary to start antibiotics quickly. And specifically for minor skin infections, it is also OK to try the less potent over the counter topical antibiotics if the individual cannot readily get to a doctor.
So, when you ask whether it can only be healed by antibiotics, the answer is no, because the body can heal many of these infections. But when you also ask if it could spread, the answer is yes, because a small percentage of untreated minor skin infections can do so.