In India as in other places, "drinks" is understood contextually.
If you generally say "he drinks" when speaking about a person, it will not be taken to mean "he drinks water". It is contextually understood you are referring to alcoholic drink regardless of if you are in India or America.
So at a pizza place in India where ostensibly no alcohol is served, you can unhesitatingly ask "What drinks do you have" and no one will assume you are asking for Old Monk.
The challenge here is not about being understood, it's about using phrasing that flows easily, is current in usage, and is not over-formal or contorted.
"Refreshments" is quite formal to the extent that it is not typically used in the context of actually ordering; it is used while planning events and menus. So if you ask for refreshments a waiter might stop and take a moment to decipher what you need.
"Soft drinks" is an outmoded term, used mostly by parents or the older generation. If you still want to be explicit, "cold drinks" is more common in the Indian context - "Do you have any cold drinks?"
"What do you have to drink?" also works, although it's so open-ended that practically I'd wonder if you expect him to recite the entire drink menu, which can be rather long at some places. At such places, just ask for the menu instead of asking for options verbally. "Can I have the menu, please?"
Lastly, in India the wine list or alcohol menu is usually a separate menu, called the "drinks menu" which you could ask for if you're at a place that serves alcohol.
EDIT: As discussed, "Cold Drinks" (or "cool drinks" in Southern India) is still used, even if it sounds old-fashioned. There is no corresponding commonly used term for hot or warm drinks, so I would personally prefer "Do you have anything warm to drink?" over "Do you have any hot drinks?". Both would be equally understood, however.