In the general public, you're correct. You have no rights over me, nor I over you. In my original post, I failed to qualify my answer - my rights extend to the limits of my property. I have *every* right to impinge upon your right to free speech in my home - I can tell you to shut up or leave. Although it gets a little fuzzier in practice, in theory, I have the right to search you without probable cause if you're on my property.
In the public forum, however, you're completely correct.
Yes, you can. But that is all you can do. You can't take legal action against me unless I refuse to take one of those two options. If I do refuse to shut up or leave, then I am trespassing and you have cause for legal action.
Likewise, if I am told my bag must be searched, I have two options: allow the search, or leave. So long as I am willing to leave the premises, there is no further action that can be taken.
Now, if I am told when I enter the store that I must consent to be searched when I make a purchase, then I give my consent simply by making that purchase. Failing to allow a search at that point means I've violated the conditions of my being there and action can be taken.
Although it gets a little fuzzier in practice, in theory, I have the right to search you without probable cause if you're on my property.
There are laws in place that give merchants the right to search in certain circumstances. The details and scope of the laws vary, but the searches can never be unreasonable because the government can't grant authority it doesn't itself possess.