This is how I integrate Google reCAPTCHA-v3 with React UI:

document.body.addEventListener("click", () => {
    e.preventDefault();
    grecaptcha.ready(function() {
      grecaptcha.execute('reCAPTCHA_site_key', {action: 'submit'}).then(function(token) {
          // Add your logic to submit to your backend server here.
      });
    });}, {once: true});

I have searched the documentation but didn't find a clear guide for integration with a React app.