This might be possible, but only if you arrange it at the very beginning.
But first, note that "Full Professor" in Europe and "Professor" in the US mean very different things. "Full Professor" is about the same in both places except that, I think, in Europe a department will have only one Full Professor and in the US there may be many. Here (US), Full professor means teaching, research, and service (with different balance different places) but no assumption of administration duties. The term Professor can apply to an Assistant Professor with a seven year probationary period leading to tenure and an Associate Professor title. Several years later, if the person has done well according to the judgement of the faculty, they may be promoted to Full Professor.
But the above is the normal path for people starting out. When a university hires a person from another institution, they are inserted somewhere in the cycle, but almost everyone undergoes a probationary period. (Einsteins excepted, of course.)
And, almost everywhere, moving from Staff Scientist to Professor would be unusual. Not impossible, but don't assume that it is a normal or common path that would obviously be open to you.
My advice would be to negotiate the terms at the very start. I would suggest something like asking to be made an Associate Professor on hire with a, say, two year probationary period. If you are a world renowned researcher (Einstein, again), then ask for more, but probably not less.
The problematic part, I think, would be that you don't say you have any teaching experience. Almost all Professors here (all ranks) do some teaching. For those doing high level research in a place in which that is the most important item, it will be less, in most cases, but usually there will be some, even if it is only advanced graduate courses. And, supervising graduate students is also an important element of the job.
If they will start you out as Assistant Professor with a long probationary period, make sure you understand the terms. How much will teaching count in a tenure decision? Research? Other? Can it be done by somebody stressing one of these more than expected, but leaving the others to wither?
But, if you don't negotiate it at the start, I fear you would get stuck with no path forward. And maybe not a path out, either.
I assume the professor you are talking to has some clout. Perhaps enough to get the university to go along with your requests. In most cases, the path I suggest (Associate Prof, short probationary period) would be palatable to some, but not all, R1 universities, which are the ones stressing research above all. After a few years, if you do well, promotion to Full Professor will open for you.
Don't be shy. Don't make assumptions.