I am not a WA attorney, but a quick check with Goigke reveals that your city just renewed its loitering law and it appears you could be arrested since the hospital is a public place. See http://clerk.ci.seattle.wa.us/~scripts/nph-brs.exe?d=CODE&s1=12A.10.100.snum.&Sect5=CODE1&Sect6=HITOFF&l=20&p=1&u=%2F~public%2Fcode1.htm&r=1&f=L3%3B1%3B12A.10.010.SNUM.
If you decide to take a stand for freedom, consult with any attorney first to be sure you do it right!
NOT LEGAL ADVICE. FOR EDUCATION AND INFORMATION ONLY. Mr. Rafter is licensed to practice before the state and federal courts in Virginia. There is no implied or actual attorney-client relationship arising from this education exchange. You should speak with an attorney licensed in your state, to whom you have provided all the facts before you take steps that may impact your legal rights. Mr. Rafter is under no obligation to answer subsequent emails or phone calls related to this matter.
Yes, you can be arrested if the police believe that you are violating the law. The arrest might later be suppressed based on constitutional grounds, but that would be for the court to decide, not the police.
I do not know if the hospital or police would have any problem with you using the King County-owned hospital in the first place, but if they did, I would advise cooperating with the police. You may have a constitutional argument about it, but that will not stop the police from arresting you.
There are many ways to challenge the law on constitutional grounds, but refusing to obey a direct police order to leave is not the way that I would choose to do it. I would cooperate with the police when they told me to leave, then I would file a lawsuit challenging the trespass notice (after carefully researching whether there is a valid constitutional claim).
It never pays to try to argue a constitutional argument with the police. They are not lawyers or judges, but they do have guns and handcuffs, and they receive a lot of protection from the law when they are attempting to enforce a statute or ordinance.