The thing is that I've never felt comfortable when eating out with acquainteces, for one thing because I'm very slow eater so I'd always left lots of food on my table, and second because I've never been at ease with talking when your mouth is full with food. Plus you cannot made dirty jokes when someone's eating :)
As things are like I know, myself eating alone and at ease,I've discovered that I really do enjoy good food and a nice drink (I was almost indifferent to it before) and that I'm not that keen on making compromises anymore when it comes to eating out, because the decision where and when to eat depends only on me. And I've also become a lot more confident when talking to complete strangers, this coming from an introvert guy.
And on top of that, people watching and listening is extraordinary!
My hypothesis as to why it feels awkward is that one is afraid that other people might think, "Oh, look at this guy, he can't find anyone to go out to eat with him". But this is stupid, because obviously those other people that are there in groups don't go out every night to dine with friends either. They might just do the eating alone nights at home by themselves. Which objectively speaking seems no much "worse" than doing it in a restaurant, except that it is more hidden from the public.
So, just realize that those other people also eat alone, but when they do they might be too scared to even go out and rather sit at home. That makes you brave and adventurous compared to them! ;-)