Ajax fail to redirect to web api

A few things to note here, first of all use the following type for script,

<script type="text/javascript">

But this is not necessary at all now, browsers are smart enough. You can always omit this and it will work.

Secondly, jQuery code needs to come inside a handler otherwise $ won't be available, what you wrote is not executed at all (you missed a () at the end). Either change your code to meet those standards or do this,

$(document).ready(function () {
    $('#btnUsingjQuery').click(function () {
             $.ajax({
                 url: "http://api.google.com/payment/widget?apikey=xxx&country=RS",
                 dataType: 'json',
                 success: function (results) {
                     window.location = 'www.google.com';
                 },
                 error: function(){
                     window.location.replace("www.google.com");
             }
 
             });
         });
});

This will attach the handler to your element. I didn't tinker with your code, which needs some modifications too.

Finally, if under any case your user is to be navigated, consider using "complete" event instead of both success and error case and navigate away from there. If it doesn't work, please check console of browser.

Make jquery function run on page load - Stack Overflow[^]
jQuery.ajax() | jQuery API Documentation[^]