Freedom of speech is already a fundamental right. It is one of the few fundamental rights that the government has been willing to recognize. That means restrictions on the freedom of expression and of the press are already subject to strict scrutiny in many cases.
However, free speech and press are not entitled to the protection of strict scrutiny in all situations. There is a large category of cases in which the courts use a less rigorous standard of review, called “intermediate scrutiny.” These cases concern situations where speech is mixed in with a large component of “conduct” and the Supreme Court has watered down the protection of speech in order to give the government a freer rein to regulate the non-speech conduct. The classic case involved a man who burned his draft card as a way to express opposition to the draft. The Supreme Court said that, in order to allow suppression of free conduct (burning the card), it would all right for Congress to suppress the expression as well.
If Americans enjoyed freedom as a fundamental right, of course, there would be no need to impinge on free speech in order to allow the government to cut down individual freedom to engage in other forms of conduct. All conduct, whether expressive or non-expressive, would have entitled to the same high degree of protection. Neither free speech nor any other form of conduct could be suppress by new or existing laws unless the laws were narrowly tailored to advance a demonstrated compelling governmental interest and were without any less restrictive alternatives.