Despite the title, I should make it clear that I oppose the proposed legislation in Arizona to allow warrant-less searches of abortion clinics. The stated justification for the law is to ensure safety and healthy conditions at clinics, but the law is transparently about harassing a particular type of business.
However, I must admit I get some schadenfreude from this. Supporters of the bill say that they are only extending the current standards applied to many other businesses, such as restaurants and bars, to abortion clinics.
Regulators from OSHA to the health department have tremendous powers to barge into private businesses and conduct searches without a warrant, whatever the text of the Fourth Amendment might say. They justify this with licensing regimes that require these businesses to have state licenses, and then require businesses accept these extra-Constitutional searches as a prerequisite for the license.
I have opposed these licensing regimes for years, in part because the consumer protection justification is often a sham -- what they really want is to be able to exercise control of private businesses. In some cases, these laws are used to protect incumbents. In some cases (e.g. here) they are used to try to shut down the entire (legal) industry.
Statists on the Left have generally poo-pooed these concerns. Their typical response is that businesses are just whining, and that only those in violation of the law have something to fear. Now, they suddenly are recognizing that an unannounced search per se is threatening.
Update: I find abortion proponents on the Left to be among the worst examples of faux libertarians. They claim their issue is about choice regarding one's body, but then tend to simultaneously support all kinds of government interventions in personal medical decision-making. They are all for the sanctity of private property when there is an abortion clinic on the site; not so much otherwise.