No. They're different.
Being selfish means that you don't care about other people. It implies that you put your needs and desires above those of others and that other people are likely to suffer because of your selfishness. Taking more than your fair share of food when others are hungry is selfish. Parking your car across two spaces is selfish.
Being self-indulgent doesn't necessarily affect other people. Nobody else suffers if you spend the day in bed eating chocolate and playing video games. If a self-indulgent journalist writes an entire article all about how sorry she feels for herself, this is self-indulgent: it's boring and even annoying to read, but it doesn't harm anyone.