This is completely true, some introverts have a natural affinity for social intelligence, but I would wager it's more likely that an introvert is socially awkward/inept simply because if social interaction makes you tired, you're less likely to actively go out of your way to engage in it, and less experienced.

To OP: I recommend the next time you're in a similar situation that you initiate yourself with some sort of buffer, even if it's something small like having something to do with your hands (operating a grill, drinking a beer) or gathering around a television, find something that you can use to occupy yourself in a space with other people so that you don't feel any pressure to fit in, but naturally have a reason to include yourself even non-socially. Hopefully overtime you will build up those "social muscles" like WulamocS said. :)