If I cover my 19 year old daughter with my medical insurance, am I responsible to pay what the insurance doesn't pay?

The general answer to this question is that the policy holder is responsible to pay that portion of the bill that the insurance company does not cover pursuant to the terms under the contract of insurance.  That is generally a co-pay is the person uses a doctor within a net work under the plan or if you have an 80/20 or some other split like that after a deductible is met, the percentage required under your plan.  So if your daughter is covered under your plan then the answer here is yes.  Please make sure, though, that the insurance is correct.  Double check everything.  Good luck.