I can't see any reason to doubt it. Soccer is an abbreviation of Association Football. Adding 'er' to a diminutive was a form of British slang in the 1920s and 1930s. The left-wing publication The New Statesman is still known as the Nagger Staggers. Other examples from the time: The Pragger Wagger (The Prince of Wales) and my own favourite The Honkers and Shankers (the Hong Kong Shanghai Bank).

WOW: GOLD. Thank-you friend.

What are the opposing theories? Similar-ish games may have come up around the world independently and stayed in isolation, but modern football/soccer definitely has its origins in late medieval Britain as a street game. In the mid-1800's, amidst the Victorian atmosphere of the time, English public (as in high end) schools saw soccer as a healthy, muscular, and moral occupation for students, and codified the rules. Modern soccer split from rugby when using hands to carry were disallowed. Soccer's subsequent spread throughout the world can then usually be traced to British/Scottish expats and coaches, as is the case with Brazil, Argentina, Austria/Germany, Hungary, etc. Inverting the Pyramid by Jonathan Wilson is a great book about the origins of soccer and its tactics.