Not in ancient Greece or Rome--as a general rule, a well off Roman going out would go to a banquet at a friend's house rather than a restaurant. There was food available to buy, but usually from small shops and probably for somewhat less well off customers.

If you are curious, I wrote something about the general dynamics of this.

The earliest real "restaurant culture" I have found is from Song Dynasty China, and it seems to have been very vibrant indeed. This probably developed somewhat out of the high class red light districts of the Tang Dynasty, but it was clearly its own thing and seems rather familiar to the experience of modern restaurants. An evocative picture of this comes from the Dongjing Meng Hua Lu, which was a sort of elegy to the splendors of Kaifeng written after its fall to the Jurchen. I unfortunately don;t have my sourcebook with me, but the Wikipedia page has a pair of choice quotations that give a good flavor. Apparently there were 72 renowned restaurants, as well as mention of numerous famous noodle places and the like.