I don't know of any states that have exit exams of the type the OP describes.

I went to school in Maryland, and we had various statewide standardized tests at various points, but I don't think you had to pass any of those. They were just for the state to assess where the students (as a body) were. I could be wrong, though. I never paid much attention to these tests because, to be quite frank, I breezed right through them and never had any concern about passing them, even if passing them were required.

To graduate high school (9th-12th grade), we needed a certain number of course credits, and in getting those credits, we had to hit certain requirements. For instance, we needed 3 years of math. I took Algebra II, Trigonometry, and Pre-calc. Others, due to what they took in middle school (6th-8th grade), started with trigonometry and/or skipped from Pre-calc and went straight from Trig to Calculus and/or went on to Advanced Placement Calculus. We needed a semester of gym, at least a year of something artsy (band, pottery, photography), 4 years of English, 3 years, etc. Because of what was on offer, most students took basically the same courses throughout. So, generally it was Earth Science-Biology-Chemistry-Physics. The only difference was whether you took the regular version of those classes, the honors version, or the AP version. If you passed the class (based on a series of assignments, not one big test or paper as seems to be common in Europe), then you got credit, and if you got all the credits, you graduated. If not, you might have to take more credits in night school or summer school, or repeat a year.

As for AP classes, those are more advanced classes, at a college level. It's a national program, and at the end you take the AP test (or at least, we were required to take it if we took the AP class). We did not have to pass the test, just take it. Our grade for the actual class itself was, like other classes, based on a series of assignments throughout the year. However, if you do pass the AP test in high school, many colleges will award you credits. Because I took several AP tests and did well on them, I started college (university) with a whole semester's worth of credit already.