The biggest sporting event on the year in the south of France is the Formula one Monaco Grand Prix, which takes place at the end of May, and is without a doubt the most glamorous event in the calendar. Many people travel to the area at that time, and Nice is a good base from which to travel to the principality in order to watch.
Nice has a football team who play in the French League, and whilst they are not the best side in the country, they enjoy healthy local support, and a visit to their ground is a great experience if you are in the city during the season.
There are plenty of golf courses nearby, and these allow non-members to play provided that you book in advance.
You can sail and windsurf from the beaches around Nice, and there are also several local companies who can arrange deep-sea fishing, and diving trips in the clear Mediterranean water.