You can almost certainly restrict it on public broadcast TV, which would be a good start. The FCC has pretty broad powers to regulate broadcast TV and radio compared to other kinds of speech. For example:

https://en.wikipedia.org/wiki/Fairness_Doctrine#Decisions_of...

What about it? I can't show pornography on TV or slander someone on a newspaper and call freedom of speech.

Legally, it's not really an issue. Commercial speech can be broadly regulated by Congress.

Free Speech doesn't really factor into a huge portion of what you see from the FDA, FTC, SEC, CFPB, and hundreds of state-level organs focused on regulating deceptive business practices.

What about it? Not all speech is protected and there is a case to be made of the harm such speech poses. This is why we can't yell "fire!" in a crowded theater. Schenck v. United States, 249 U.S. 47 (1919)

The bar for prior restraint based on content is pretty high. I believe it's "clear and present danger". You're likely to have a difficult time making the case that ads for allergy medications present a clear and present danger.

The bar for prior restraint on public airwaves is much lower. We'd be much better off if drug marketing were relegated to the shady corners of the internet for advertising.

I agree - this could be tricky.

One solution for restricting advertising would be to allow consumers to sue drug companies for false claims or for misinforming doctors.

No, it's commercial speech. The FDA already regulates what they can say in drug advertisements. There's not really a legal issue there.