I have no doubt that the President and Federal government maintain
jurisdiction over federal lands, but it is less clear to me the
jurisdiction the Federal government has over private or state lands.
The authority of the President is the authority delegated to the President to promulgate regulations under federal environmental laws and laws governing the management of federal lands.
Congress clearly has jurisdiction and authority to enact federal laws that impose environmental regulations upon both private land and upon state owned land. And, it has done so. Among the more notable are the Clear Air Act and the Clean Water Act. There are also multiple environmental laws governing the pollution of land with various kinds of pollutants. All of these laws apply to privately owned land and some apply to state owned lands subject to certain exceptions which are not complete exemptions for all state owned property.
Congress has also enacted industry specific regulations of the oil and gas industry that are not exclusively environmental laws. And, Congress has enacted laws setting guidelines regarding how federally owned lands are managed. The Bureau of Land Management, the Forest Service, and the National Park Service are among them.
In particular, the federal government reserved the mineral rights in most, but not all of the Continental U.S. to the West of the Continental Divide, at the time that the surface rights were homesteaded or otherwise transferred from federal government ownership to private ownership. The story of how it got that way, which has several parts and is rather complex, is retold in a law review article.
This needs to be compared to the places where there are oil and gas resources suitable for fracking, roughly speaking, Texas, Oklahoma, Louisiana, Northern Michigan, the Appalachian mountains, select places near the Rocky Mountains, and isolated other sites.
So, a very large share of all oil and gas operations that take place on land where the surface rights are owned by private persons or state governments are actually exploiting federal owned mineral rights. In this capacity, the federal government has the same rights that a private owner of the mineral rights would, i.e. the right to lease the mineral rights to ah oil and gas operator who would then extract the minerals pursuant to that lease in exchange for a percentage of gross revenues generated and other limitations in the lease.
The President can't revoke existing oil and gas leases, but can decline to issue new fracking leases for federally owned mineral rights (regardless of who owns the surface rights to the land in question) and can established new environmental regulations that limit fracking, often simply by rescinding new Trump administration regulations that, for example, allowed oil and gas leasing closer to National Monuments and National Parks than had previously been permitted.
The President could also toughen the regulations governing the cleaning up and shutting down of oil and gas extraction sites in a way that makes it less economic to frack, could strengthen OSHA regulation of oil and gas worker safety, and could authorize states to issue their own environmental regulations with respect to private land that would otherwise have been pre-empted by federal law - something the Trump Administration rescinded permission to do in Oklahoma with respect to land (almost half of the state) that the U.S. Supreme Court recently declared to be part of an Indian Reservation.
To say that that the President can or will "ban fracking" oversimplifies the matter and amounts to hyperbole. But the President's regulatory power over the oil and gas fracking industry that can be exercised without Congressional action is considerable.
Also, when a President promises to do something if elected, there is often an implied notion that some of this will be done by pushing for legislation in Congress to give the President this power and Congress certainly has the authority to give this power to the President.