It must be very frustrating to have a painful infection in such a tender area. You are right that having a piercing in the navel does make it much more prone to infection. However, our skin is covered with bacteria which means that it is always possible to get an infection through any sort of microtrauma or abrasion to the skin surface. You may have developed some friction at that site from a piece of clothing (such as a belt) that allowed some bacteria to get under the surface of the skin and cause a cellulitis, or skin infection. It is also possible that you may have something other than an infection that is causing the pain and redness in the area.
Overall, to determine whether this is an infection that requires antibiotics or whether there is something else going on, it is definitely a good idea to see your primary care physician
right away. If you start developing fevers or chills or notice pus coming from the area around your navel, you should be seem immediately, either in a walk-in urgent care facility or in the Emergency Department. Hopefully this is a minor infection that can be treated with oral antibiotics. However, you do want to have a physician examine you to make sure this is not something more serious.