Kind off. You don't need to spend the tax money helping the environment, you just need to restitute whatever utility those affected by the negative externality have lost. For example, people may be willing to accept an increase in CO2 in exchange of a direct transfer of $100 per ton emitted.

Assuming that tax money is being used effectively (lol), the negative externality could completely disappear with the right tax level. You could even turn driving into a positive externality if the tax is high enough, while still polluting.

Well, it's being paid. Maybe it's not being used well, but it's being paid.