TOKYO—Democracies in Asia that rely on the backstop of U.S. military power for their prosperity are confronting a new reality: American protection is no longer enough now that China rivals the U.S. in areas such as advanced missiles and naval hardware.

To tackle the problem, Beijing’s neighbors, with prodding from the U.S. and help from Europe, are building a network of regional security ties with a goal similar to that of the North Atlantic Treaty Organization: deterring a large nation whose growing ambitions have raised the prospect of conflict.