If you check out http://recaptcha.net/apidocs/captcha/client.html it says:
"In order to avoid getting browser warnings, if you use reCAPTCHA on an SSL site, you should replace http://api.recaptcha.net with https://api-secure.recaptcha.net."
So clearly recaptcha supports HTTPS submissions. Does the ASP.NET control have any properties you can configure the outbound URL? At worst you might need to use Reflector to examine the code and see how it's built.