No you won't damage your car if you just drive short distances since cars and other vehicles are made to drive both short distances and long distances.
However while short distance driving will not damage your car it's not the best thing for your car either to just drive short distances.
If you just drive short distances then your engine doesn't heat up enough to keep the moisture out of the engine oil so you may get condensation in your engine oil which is normal but driving the engine long distances will heat the engine up and engine oil up enough to evaporate the moisture in the engine oil.
So if you just drive short distances then the engine won't heat up good enough so you should take the vehicle out at highway speeds at least once per month or more often to bring the engine and engine oil up to operating temperature.
Engines that are up to a higher temperature due to highway driving usually last longer than vehicles that just drive short distances.