No, no it does not. Here's the actual text of the first amendment:

"Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the Government for a redress of grievances."

So basically, among other things mentioned in the amendment, the U.S. government can't make laws that prevent people from speaking their mind (in the U.S.). How does this translate to "people can annoy people online with impunity"?

This meme was inspired by people on forums whining about having posts deleted or being banned because of things they say. They seem to believe that "free speech" means they can say whatever they want, wherever they want, and no one can stop them. That is false.

XKCD made a comic about this. Go ahead and Google "XKCD free speech".