California Penal Code 647f states that being intoxicated in public is prohibited. When the police arrived, they were confronted with probable cause for an arrest. They (presumably) became aware of the matter because the doctor called the police, since she believe that you would drive drunk. (We can inquire into whether that was a reasonable belief, but it doesn't matter, what matters is that she had the belief and acted on it). Now the question is whether the doctor acting on the belief (making the call) was legal. A negative answer does not affect the legality of the arrest.
There is also a law imposing on medical professionals a duty to report, which is fairly wordy, but does not seem to directly require reporting the fact that a person is publicly intoxicated. However, attending circumstances could have suggested one of the triggering causes for mandatory reporting (wounds, for example). Again, it does not matter (to a point) if, in the light of close scrutiny, the doctor's conclusions were mistaken. When doctors are required to report facts to the police, reasonable over-reporting is not penalized. There is also no law against calling 911 to report a potential DUI (the usual public-campaign focus is on those actually driving). So calling the police under the circumstances falls between "allowed" and "required".
The HIPAA privacy rule could be relevant because that theoretically could block the doctor from making the call. (Note that the doctor, and not the patient, is bound by the confidentiality requirements). §160.203 allows exceptions to the confidentiality requirement if "necessary... For purposes of serving a compelling need related to public health, safety, or welfare", so an exception may have been granted. If this was done within the scope of a mandatory reporting law, it is legal to disclose PHI; under §164.512 it is allowed, "to prevent or lessen a serious and imminent threat to the health or safety of a person or the public".
A confidentiality agreement would not increase your chances of being arrested. If the doctor's confidentiality statement were less restrictive than HIPAA, HIPAA prevails (the law trumps contract terms). If it is the same as HIPAA, it has no effect (and simply states what HIPAA says – the normal case). If the agreement were more restrictive, it is possible that the doctor calling the police would be a breach of contract, unless the call was required by law. You would have to see what in the agreement would have prohibited calling the police. But that would not affect the validity of the arrest.
To re-phrase the matter: the arrest was because you were found to be intoxicated in public. The police were there and could judge your state (probable cause). They were there by permission of the property owner, so the arrest was not unlawful for lack of a warrant. That is as far as one can go in searching for an illegality to the arrest itself. One might go further and ask whether the doctor has committed an actionable wrong by calling the police with her suspicions. This could go either way: it really depends on the full set of details, regarding your condition. If the doctor suspected that your actions fell under one of the mandatory reporting categories, she had to report, and otherwise it is not prohibited under HIPAA.
If a person is intoxicated and answers the question "Would you normally proceed to drive home in this state?" in the affirmative, then it is a reasonable inference that the person will do so. An answer "No, absolutely not", on the other hand would work against the "public danger" inference: that has no effect on the arrest, but could have an effect in a suit against the doctor (violation of the privacy rule). In such a suit, the doctor's defense would presumably be that despite the answer, she still had a reasonable belief that you were a public danger. Then the matter would reduce to what other facts she knew of that would support a public danger conclusion.
