They can regulate whatever they want. Tomorrow they might pass a law that makes it illegal to display the Google logo. That has no impact on Google's rights.

In some countries, women are not allowed to leave the house unless accompanied by a man. We recognize that women do have the right to do so, and that countries that disagree are stupid and wrong.

You're talking about moral rights here, not legal rights. Moral is highly subjective, you know.

Of course I am. That's the only kind of right it is possible for non-lawyers to have a discussion about, since legal rights either exist or don't.

"Government can regulate what content can and cannot be shown"

Depends on the government. In the US, commercial speech (IE speech for the purposes of engaging in commerce or advertising) is a weird issue. Content-based regulations of commercial speech have to satisfy intermediate scrutiny (instead of strict scrutiny that noncommercial speech would have to satisfy).

Intermediate scrutiny is an odd test, but suffice to say, the government is not going to win in a lot of cases.

Essentially, at least in the US, the government's authority to restrict commercial speech has significant limits, particularly when it comes to content-based regulations.

> Government can regulate what content can and cannot be shown (e.g. "The right to be forgotten")

Sure, and "hate speech" laws and illegal materials.

But they cannot regulate you, for instance, into showing something that you are not showing.

Suppose I run a search engine and some politician is against my organization. I can drop most positive entries related to that politician from my index, keeping mostly the ones that point to negative material.

they can, and they did, for example, for the browser ballot in windows.

This was not a law, and so a very different situation. This was a court ordered remedy for a conviction.

Courts generally have the power to fashion appropriate remedies, and yes, sometimes they involve compelling speech or other things.

(Some of these remedies have been struck down depending on what they have been forced to say or not say).

Warning stickers on cigarettes are mandated by law.

That is a labeling requirement for health, not compelled commercial speech (which again, is about commerce :P). It does not require them to state anything about themselves, just about the risks of the ingredients in cigarettes.

You need to understand these are all very different things in the eyes of the law

Also note the

A. current warning labels were litigated, and tobacco won at the district court level: http://www.news-record.com/content/2012/03/01/article/judge_...

This order was reversed by a divided panel on appeal: http://www.journalnow.com/business/business_news/local/artic...

The supreme court declined to hear the case (which means nothing :P)

B. Requiring phillip morris, et al to make ads stating they deceived the public was not okay. http://www.claimsjournal.com/news/national/2015/05/22/263534...

etc

These are all very complex constitutional free speech issues, and trying to simplify them like you have with a throwaway line doesn't work well in practice.