If I put in unleaded gas into a 2004 Volkswagen Jetta VR6 would that severely damage the car?

No. It requires unleaded.

If you put in leaded the only thing it would hurt was the catalytic converter. There is no on-highway leaded gas anymore--it's illegal to sell it. Aviation gasoline is leaded, as is racing fuel, but not road gas. The VR6 has two knock sensors on it. Knock, which is caused by running too low octane, trashes engines, and the knock sensors are there to retard the spark and get rid of knock. You won't hurt the engine by putting regular unleaded in it, but it won't run as well as it could with more-expensive gas in it.