Yes, he could. But the Executive Branch would still have the responsibility to enforce the laws that caused the creation of the EPA in the first place:
Air
1955: Air Pollution Control Act PL 84-159
1963: Clean Air Act PL 88-206
1965: Motor Vehicle Air Pollution Control Act PL 89-272
1966: Clean Air Act Amendments PL 89-675
1967: Air Quality Act PL 90-148
Water
1948: Water Pollution Control Act PL 80-845
1965: Water Quality Act PL 89-234
1966: Clean Waters Restoration Act PL 89-753
1970: Water Quality Improvement Act PL 91-224
Land
1964: Wilderness Act PL 88-577
1968: Wild and Scenic Rivers Act PL 90-542
1970: Wilderness Act PL 91-504
Endangered species
1946: Fish and Wildlife Coordination Act PL 79-732
1966: Endangered Species Preservation Act PL 89-669
1969: Endangered Species Conservation Act PL 91-135
Hazardous waste
1965: Solid Waste Disposal Act PL 89-272
1970: Resource Recovery Act PL 91-512
Other
1947: Federal Insecticide, Fungicide, and Rodenticide Act PL 80-104
1969: National Environmental Policy Act PL 91-190
Nixon created the EPA because competing bureaucracies at Agriculture, Interior, Justice, Commerce, HEW and Labor were falling over each other trying to enforce this incoherent mishmash of unconstitutional laws. Eliminate the EPA and you will just return to the status quo before 1972, and all the watermelons from EPA will be hired by those same cabinet departments listed above and start competing with each other to see who is the most radical.
But the Executive Branch would still have the responsibility to enforce the laws that caused the creation of the EPA in the first place:
Only those LAWS he Agrees With, See Barry and Immigration.