Does a professor have the academic discretion to design the curriculum for his class?

Answering for the R1 schools with which I am familiar:

Of course a professor has this discretion. There is no universal standards board that decides what must be taught in what course -- nor would we want such a thing. Otherwise we might end up teaching intelligent design alongside evolutionary biology and the great flood alongside plate tectonics.

The professor might be better off choosing a different course title, but there are a host of bureaucratic reasons why this doesn't always happen. So long as the syllabus accurately reflects the course material, I see no problem whatsoever. (And even if it doesn't, the professor still has the ultimate right to determine the course curriculum).

In addition to the question of what a professor is permitted to do, there is a second question about what is a reasonable standard of conduct for the professor. This too offers quite a lot of latitude, particularly in higher level classes.

Caveats: For certain large introductory classes or multi-selection courses, the academic department or a committee within the department may select a set of topics to be covered in order to standardize what the students learn -- but in this case the professor would not be tasked with developing the curriculum himself or herself. As we move from introductory classes toward advanced classes. Similarly there could potentially be a problem if a professor taught oceanography in an English literature class, or vice versa -- but decisions about where to fall on the philosophy / calculation spectrum within a given topic area are entirely up to the professor.

Addendum This whole thing may not seem like a big deal to those of us in STEM fields (with the possible exception of those fields that sometimes contradict big oil, big tobacco, or a narrow reading of the Book of Genesis). But faculty discretion over the contents of the curriculum is extremely important the humanities and social sciences where political agendas are more immediate and where the choice of a text may itself be an overtly political act.