I'm going to edit the practice area of your post to "Debt Collection" for better exposure.
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Yes, you owe the debt. You received the services. The police were legally obligated to take you in for examination based upon their belief you were a threat to yourself. You can question whether that belief was reasonable, but the police will say it was, and the only way to get past that is to sue them for violating your civil rights, which will be a long, expensive, and probably unsuccessful case. In the meantime, you'll still owe the hospital bill - the cops aren't going to pay it and, legally, they aren't liable for it since you are the one that received the services.
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