From a UX perspective, it can help to visibly let the user know when they can proceed to submit the form - either by enabling a disabled button, or simply making the button visible.
Here's a simple example...
<form>
<div class="g-recaptcha" data-sitekey="YOUR_PRIVATE_KEY" data-callback="recaptchaCallback"></div>
<button type="submit" class="btn btn-default hidden" id="btnSubmit">Submit</button>
</form>
<script>
function recaptchaCallback() {
var btnSubmit = document.getElementById("btnSubmit");
if ( btnSubmit.classList.contains("hidden") ) {
btnSubmit.classList.remove("hidden");
btnSubmitclassList.add("show");
}
}
</script>