Valley Medical Center is in violation of state preemption. They are not a state agency, nor are they an area specified in the RCW that prohibits weapons. They are, however, a public entity.
Cities, counties and municipalities cannot create rules to prohibit firearms in areas not specified in the RCW. Seattle tried this with the parks gun ban and lost in court. Currently under appeal, they will lose again, especially now that the 2nd Amemndment comes into play, when it did not before.
State agencies create rules, called WAC's, that govern how they do things. Yes, they can require you to leave your gun in your car. Is there a criminal penalty if you violate this? Dunno, read the applicable WAC for the agency you have a question about.
Reading the WAC's that pertain to Colleges and Universities, I can find no criminal penalties, unless they tell you to leave and you refuse, then criminal trespass comes into play.
Most College and University WAC's that prohibit firearms lsit the sanctions that can be taken against students and faculty/staff. Ranging from expulsion to termination. others can only be required to leave.
The WAC that governs this area for the University of Washington, interestingly, specifies that you must be asked to leave by a "uniformed officer from the UW Police Department". Failure to leave would then result in arrest and charges of criminal trespass.