Stability of Nitromethane
It depends on what kind of reactivity you mean. Nitromethane is indeed flammable and is used as a racing fuel. It can also detonate, especially when mixed with an oxidizer. But it is generally quite stable to atmospheric oxygen at room temperature, perhaps because the nitro group is electron-withdrawing and reduces the likelihood that the hydrogens on the methyl group will react with oxygen or other oxidizers.

In other ways nitromethane is indeed reactive: for example, the nitro group makes the hydrogens on the methyl group acidic, so it reacts with an aqueous base like sodium hydroxide to give the nitromethide anion, which is useful in organic synthesis.

Consider the splitting of nitromethane into methyl radical and nitrogen dioxide. It has a normal sigma bond between carbon and nitrogen, and to break that bond you have to put a lot of energy in. You are right that NO2 is indeed stable on its own, but it is still a reactive, high energy species. Methyl radical is even more reactive, and the energetics of this bond-breaking reaction are overall unfavorable, at least at lower temperatures. Higher temperatures, such as might occur when sparking a detonation, may give the kind of splitting you are talking about, plus a lot of other reactions as well.