What percentage of a professor's salary is paid for by tuition? Is it possible to determine what percentage of a tenure track/tenured professor's salary is paid for by tuition? At my UK university, our school budget includes income from student fees (we only get a portion of the fees each student pays), grants, and central university funds (presumably some of this is indirectly from student fees). I am curious if a calculation could be made for STEM departments at US R1 type universities. Presumably some of this information may be available for state schools.

I think the answer I am looking for would be something along the lines of the ratio of the total amount of tuition fees (hopefully divided up into undergraduate and graduate fees) given to the department divided by the total costs of teaching (again, ideally divided up into undergraduate and graduate costs). In other words if a department gets $500,000 from tuition fees and the teaching costs are $1,000,000, then 50% of the teaching costs are paid for by tuition. Teaching costs would have to include space charges, IT charges, printing charges, and the cost of staff time. These all seem to be known, well defined, quantities. An exact estimate of the cost of staff time would require going through each member of staff and prorating the salary by the percentage of time allocated for teaching (e.g., 0% for someone who has bought out his teaching and maybe 40% for someone who has not). Presumably, someone in the university/department has access to these numbers.

This is only a guess as to what an answer will include, but if there is a different (more general) way of getting to the answer, that is fine too.
I think that you'll never be able to sort this out. Most universities cross-subsidize departments to the point that even individual departments don't know how much of their own faculty lines come from the tuition of students majoring in that subject vs. tuition from other departments' students. Does the English department fully fund itself through the fees students pay for English classes? I doubt it. State funding, though a diminishing portion of most state universities' budgets, picks up some fraction of the costs. 

Also, departments see level funding in the face of modest fluctuations in class enrollments. If 45 students took Calculus I last fall and 52 sign up this fall, the department doesn't see increased revenue for that. The department has a number of faculty lines coming from the dean's office to teach the projected enrollment. If that spikes or drops off substantially such that more instructors are required, then the department has to negotiate more or fewer lines with their dean. 

That being said, I think that most departments negotiate expected breakdowns in effort with the faculty they hire. Of a typical 9 month appointment, a faculty member might be expected to do 35% teaching, 35%  service, and 30% research (or whatever), and that might be the expectation regardless of whether the teaching load is a 1-1, 2-1, or 2-2. Where, again, it's entirely unclear whether that 35% teaching is completely funded by tuition or partly funded by tuition, central endowment, and/or a local named chair.