UPDATE 2018: AndroidJSCore has been superseded by LiquidCore, which is based on V8. Not only does it include the V8 engine, but all of Node.js is available as well.
You can execute JavaScript without a WebView. You can use AndroidJSCore. Here is a quick example how you might do it:
HttpClient client = new DefaultHttpClient();
HttpGet request = new HttpGet("http://your_website_here/file.js");
HttpResponse response = client.execute(request);
String js = EntityUtils.toString(response.getEntity());
JSContext context = new JSContext();
context.evaluateScript(js);
context.evaluateScript("question.vote(0);");
However, this most likely won't work outside of a WebView, because I presume you are not only relying on JavaScript, but AJAX, which is not part of pure JavaScript. It requires a browser implementation.
Is there a reason you don't use a hidden WebView and simply inject your code?
// Create a WebView and load a page that includes your JS file
webView.evaluateJavascript("question.vote(0);", null);