There shouldn't be an issue with https, that's all my site allows. But there could be an issue if your call to bring in recaptcha isn't also using https.
Some browsers won't bring in http resources for a page that is https.
I use this, but I'm using explicit rendering, so your api call probably looks a little different.
Another thing you can do is just not specify the url schem - then the browser will use the protocol that was used to bring up the page. It looks odd but it is allowed and works:
If someone is accessing your site with http, then that call will use http but if they are accessing your site with https, then that call will use https. It won't work if you're just bringing the page up from the file system for testing though because then it tries to use a file:/ scheme which can't get out to the server.
-Seth