The general claim people make suggesting things like Right To Be Forgotten or Right To Privacy is that it violates "free speech", and particularly, that via Citizens United, the US currently claims corporations have the right of free speech. Google believes, for instance, all limitations on how it provides search results as an infringement on Google's right to "say" whatever they want.
Of course, we already have a variety of limitations on free speech, plenty of laws that restrict what companies and and cannot disclose about other entities, and GDPR would be no different.
> [a] Right To Privacy is that it violates "free speech"
Copyright is settled law that definitely limits the kinds of speech people can engage in.
Couldn't you construct a right to privacy by first saying an individual has an automatic ownership right to certain kinds of personal data [1], and that data can't be used without an appropriately constructed license [2]?
[1] You might be able make this strong and compatible with the First Amendment, by treating machine-collected and person-collected data differently. A machine has no First Amendment right to speak about what it knows, while a person does.
[2] The "appropriately constructed license" requirements could include GDPR-like definitions of consent.
I, personally, do not believe there is any disagreement between the Constitution and GDPR. In fact, a right to privacy has long been inferred by combining traits of a few amendments: https://en.wikipedia.org/wiki/Right_to_privacy#United_States (Of course, that constructed right is a right to privacy specifically from government actions.) And generally, we've recognized that some rights such as speech, may need to have limits to avoid encroaching upon other rights, such as privacy.
Of course, we already have a variety of limitations on free speech, plenty of laws that restrict what companies and and cannot disclose about other entities, and GDPR would be no different.