// https://github.com/angular/angular.js/blob/d606e66f7ab893f8799991883fd02b0e66e3225e/src/ng/directive/input.js#L2094
// のrequired判定する部分だけ拝借します。
angular.module('plunker', []).directive('silentRequired', function() {
  return {
    require: 'ngModel',
    link: function($scope, $element, $attrs, ctrl) {
      var validator = function(value) {
        if (ctrl.$isEmpty(value)) {
          ctrl.$setValidity('required', false);
          return;
        } else {
          ctrl.$setValidity('required', true);
          return value;
        }
      };
      ctrl.$formatters.push(validator);
      ctrl.$parsers.push(validator);
    }
  };
});
<!DOCTYPE html>
<html ng-app="plunker">

  <head>
    <meta charset="utf-8" />
    <title>AngularJS Plunker</title>
    <script>document.write('<base href="' + document.location + '" />');</script>
    <link rel="stylesheet" href="style.css" />
    <script data-require="angular.js@1.2.x" src="https://ajax.googleapis.com/ajax/libs/angularjs/1.2.17/angular.min.js" data-semver="1.2.17"></script>
    <script src="app.js"></script>
  </head>

  <body>
    <div>普通</div>
    <form name="form1">
      <input type="text" name="input1" ng-model="input1" required>
      <span ng-show="form1.input1.$error.required">何も入力されていません</span>
      <button>submit</button>
    </form>
    <br />
    <div>ポップアップしない</div>
    <form name="form2">
      <input type="text" name="input2" ng-model="input2" silent-required>
      <span ng-show="form2.input2.$error.required">何も入力されていません</span>
      <button>submit</button>
    </form>
  </body>

</html>
/* Put your css in here */