I've read most of Mahan's books and I've read at least some of Zeihan's books. Mahan's view is essentially that battle fleet dominance combined with ownership of key naval strongpoints (think Gibraltar) makes military success and overall dominance more likely.
Zeihan's view diverges somewhat from Mahan's points, in part because he has different views, and partly just because we're in a different era than the one Mahan was writing for. Mahan earned a lot of credibility because the late 19th century and early to mid 20th century was one in which there could be decisive naval battles, and a powerful battlefleet could virtually guarantee sea dominance.
Mahanism isn't really a prophecy so much as it was a political program pursued by Mahan himself and his allies in the legislature, the presidency under Roosevelt, the business community of his time, and with the general cooperation of the British. German naval leaders during WWI and WWII tried to implement Mahanism as if it were an ideology, but failed in part due to German naval geography and in part because they just were never able to get the level of political support that would have been necessary to rival the Atlantic powers on water. So, at least in the US, it wasn't "self-fulfilling" -- he made it his life's work to fulfill it. At a time in which most of our foreign policy intellectuals fail at accomplishing their goals, it's worth noting that he succeeded, regardless of whether or not one thinks those goals are good.
The era that we are in now is one in which coastal defenses are more powerful and unpredictable than they have been in the past. The role of aviation and aircraft carriers is also highly debatable until we have a real peer to peer war with nukes.
Personally, I found Zeihan a bit exasperating in that his analysis is often good, but he's also often glib and breezy, particularly on analysis of Russia, China, and some other countries. But he's very good at talking about some of these crucial big picture topics that get taken for granted to a popular audience.