Britons voted by 51.9 percent to quit the 28-country union, shocking markets that had priced in a win for the remain camp.
"This is the worst period I recall since I've been in public service," Greenspan said on "Squawk on the Street."
"There's nothing like it, including the crisis — remember October the 19th, 1987, when the Dow went down by a record amount, 23 percent? That I thought was the bottom of all potential problems. This has a corrosive effect which is not easy to go away."
(You can watch the full broadcast interview on CNBC PRO.)
The former Fed chairman said that the root of the "British problem is far more widespread." He said the result of the referendum will "almost surely" lead to the Scottish National Party trying to "resurrect Scottish Independence."
Greenspan said the "euro currency is the immediate problem." While the euro and the euro zone were major steps in a movement toward European political integration, "it's failing," he said.
"Brexit is not the end of the set of problems, which I always thought were going to start with the euro because the euro is a very serious problem in that the southern part of the euro zone is being funded by the northern part and the European Central Bank," Greenspan said.
Even with that in mind, the European Central Bank is limited in what it can do because these fundamental problems like the stagnation of real incomes don't have easy solutions, Greenspan told CNBC.
"There's a certain amount that monetary policy can do, but our problem is fundamentally fiscal," he said, adding that this is true in the United States as well as "every major country in Europe."
Part of the problem is that the "developed countries are all aging very rapidly," which is leading to a higher ratio of government spending in the form of entitlements, Greenspan said.
The 90-year-old Greenspan presided over the Federal Reserve for 19 years, starting with the administration of President Ronald Reagan through that of George W. Bush.