No. All chemical reactions have an activation energy; that is, there is an energetic barrier to the reaction and you have to put in some energy to overcome this barrier. Thermal energy at room temperature would not be sufficient to get hydrogen and oxygen to react, but if you added a spark, you could overcome the barrier and get them to react.
The situation is analogous to starting a fire. For a fire, you need fuel (hydrogen), air (oxygen), and an ignition source (something like a spark to provide the necessary activation energy). This is also why things like gasoline and wood do not spontaneously combust.