Platforms like Facebook would be well within their rights to try and prevent politically targeted advertising, even if it would be a fool's errand. Outlawing it would be unconstitutional.
If they try to prevent "spreading of false information" by political advertisers I've no doubt they will simply be harsher on the propagandists who have a political aim at odds with Facebook's interests, one of which is stopping these hit pieces by those angry that Trump won.
Nobody would be talking about this if Cambridge Analytica worked with Hillary. They simply want to stop their opponents from using the useful tool that is targeted advertising.
Outlawing political advertising is not what I propose. I believe propaganda is different in its intent: in my opinion direct disinformation or dishonesty (with intent) would be a sufficiently high barrier, given that it would require a high barrier of proof that the supporters were seeking to manipulate opinion with lies. This would ensure nobody would be prosecuted except in the most egregious of cases. I also believe this could be a valid exception to the first amendment in the same vein as libel or slander: spreading false information, with intent, possibly for personal gain. The parallels certainly exist.
Furthermore, I believe I would be just outraged if Hillary did this, and I think this is a pointless distraction. I didn't vote for her and I know that she also had her own shady internet propagandists working too. I think we should do our best to make sure political discussions happen organically, from real people.
It's literally the first amendment.