The long answer is...no, a BS in EHS does not have ANYTHING that will meet pre requisites for medical school other than the core curriculum, which is contained in nearly every other degree.
But neither does a BA in history or any number of other liberal arts degrees, but students with those majors get into medical school every year.
The only reason a BS in EHS would have any of the pre-requisite coursework for medical school contained in it would be because you chose to add it as an elective.
The bottom line is you will need, at a minimum, a bachelors degree in SOMETHING, and then the pre-requisite coursework, as others have pointed out.
The pre-requisites vary by school, however, the typical pre-requisites common among nearly all medical schools are:
2 semesters of major's biology with lab (not allied health, which is likely what you got in paramedic school)
2 semesters of major's chemistry with lab (again, not allied health)
2 semesters of organic chemistry with lab
2 semesters of physics with lab (can be trig or calculus based for the most part)
Along with those pre-requisites, many schools require genetics and a statistics course. Biochemistry is sometimes seen as a requirement too.
You can get all of those pre-requisites after you have gotten your degree in EHS, or you can pursue a degree in whatever and throw those courses as electives or "extra" coursework and meet the pre-requisites while obtaining your degree.
Believe it or not, there are actually some schools that do not even require a bachelor's degree. The odds of getting into one of these school sans a bachelor's degree are virtually zero, but a bachelor's is not actually a requirement in some areas.