Okay, so based on user3153169's suggestion to try vclick, I used jquery mobile to fix this issue. Note that this doesn't solve my real-world problem (the introduction of jquery mobile causes issues of its own), but it does work in this simple instance.
Here is the working code:
<!doctype html>
<html>
<head>
<meta charset="utf-8">
<meta http-equiv="content-type" content="text/html;charset=utf-8" />
<meta http-equiv="X-UA-Compatible" content="IE=edge">
<meta name="viewport" content="width=device-width, initial-scale=1">
</head>
<body>
<div id="test" style="display: block; float: left; height: 50px; width: 50px; background-color: blue;"></div>
<script src='//ajax.googleapis.com/ajax/libs/jquery/1.9.0/jquery.min.js'></script>
<script src='//ajax.googleapis.com/ajax/libs/jqueryui/1.9.2/jquery-ui.min.js'></script>
<script>window.jQuery || document.write('<script src="lib/jquery/jquery-1.9.0.min.js"><\/script>')</script>
<script>window.jQuery.ui || document.write('<script src="lib/jquery/jquery-ui-1.9.2.min.js"><\/script>') </script>
<script src='lib/jquery/jquery.mobile-1.4.2.min.js'></script>
<script src='lib/jquery/jquery.ui.touch-punch.min.js'></script>
<script>
$("#test").draggable();
$("#test").bind('vclick', function () {
alert("vclick!");
});
</script>