From an SEO point of view, no, it's not a problem for your page to have no root. Search engine's index URLs, it doesn't matter if they're in subfolders or not.
However, from a user-experience point of view you really shouldn't assume that anybody who lands on the root of your site wants to see the English version. Let's say a Japanese user tells one of their friends about your great site at
www.example.com and the Japanese user types it into their browser. They're then forwarded to an English version of your site, even though they can't read English. Most likely they will simply navigate away, and never think of your website again.
An appropriate solution would be to attempt to detect the browser's language server-side, and then redirect to the appropriate language. Failing that (if the language isn't set, or if there's multiple languages), make the root page something where the user is simply asked to choose their preferred language and then redirected to the appropriate homepage.
You probably wouldn't be hurt by simply redirecting to the English version, but your non-English speaking users will likely have a better experience if my method above is followed.