How much structural support does a Gazebo require?
Get an architecture plan and take it to the city for a permit.  They will tell you if the supports are sufficient.  Despite the obvious statement that you should always get a permit, you should definitely get a permit for anything built outside because otherwise a complaint from a neighbor can force you to tear it down.

Supporting the roof is not a problem.  Supporting the weight of all the people that could fit inside it is a problem.  I have a 12'x14' deck that required 10 concrete posts driven 5' deep (the frost line where I live in Canada) in order to be able to support a full load of people that could possibly fit during a big party or something.  It would have been 15 posts if it wasn't bolted to the house foundation.  I could hold an elephant on it, but that's what the city demanded (and rightly so).

Note:  Wood should not be driven into the ground.  It'll rot.  Either float it on concrete pads or put concrete posts down to below the frost line (if any).