No they don't.
At the core, the Olympic Games of the Olympiad (summer games) require about 72-80 nations to be recognized as an international sport and they must have held a World Championship or two for inclusion, by the IOC, into the Olympics. For the Winter Games, the standard is about 40 nations.
The only place where the country and host city has any leeway is in the cultural program. That program has included official and unofficial demostration sports in the past. This has included the following:
1. Tennis and Baseball at the 1984 LA Games
2. Curling at the 1988 Calgary Winter Games
3. Sumo Wrestling at the 1998 Nagano Winter Games
4. Rodeo at the 2002 Salt Lake Winter Games
5. Cheerleading at the 2006 Torino Winter Games
For more info please check the olympic charter which I have in a pdf link below.