This would be a very foolish thing to do. Both Chrome and Firefox, along with many other web user agents, will speculatively fetch (GET) some or all of the links on a page, including your request_method=DELETE
URLs. You will find your database has been emptied out just because someone was looking around. Do not deliberately break HTTP. GET is defined to be a "safe" method, meaning it's okay to GET any URL you like and nothing bad will happen.
EDIT for others in similar situations:
The OP says he is using JSONP and is in control of both the API server and the client web app. In such a case the ideal solution is Cross-Origin Resource Sharing (CORS, spec), although this technology requires IE8+, Firefox 3.5+, Safari 4+ or Chrome 3+. If you need to target earlier browsers, and you control both domains, I would recommend merging the content of the two domains at least for your own client web app. The api domain can remain for external clients, but they would be restricted by the CORS browser requirements.