Solution 1:

This function opens a URL without requiring additional user interaction.

/**
 * Open a URL in a new tab.
 */
function openUrl( url ){
  var html = HtmlService.createHtmlOutput('<html><script>'
  +'window.close = function(){window.setTimeout(function(){google.script.host.close()},9)};'
  +'var a = document.createElement("a"); a.href="'+url+'"; a.target="_blank";'
  +'if(document.createEvent){'
  +'  var event=document.createEvent("MouseEvents");'
  +'  if(navigator.userAgent.toLowerCase().indexOf("firefox")>-1){window.document.body.append(a)}'                          
  +'  event.initEvent("click",true,true); a.dispatchEvent(event);'
  +'}else{ a.click() }'
  +'close();'
  +'</script>'
  // Offer URL as clickable link in case above code fails.
  +'<body style="word-break:break-word;font-family:sans-serif;">Failed to open automatically. <a href="'+url+'" target="_blank" onclick="window.close()">Click here to proceed</a>.</body>'
  +'<script>google.script.host.setHeight(40);google.script.host.setWidth(410)</script>'
  +'</html>')
  .setWidth( 90 ).setHeight( 1 );
  SpreadsheetApp.getUi().showModalDialog( html, "Opening ..." );
}

This method works by creating a temporary dialog box, so it will not work in contexts where the UI service is not accessible, such as the script editor or a custom G Sheets formula.

Solution 2:

You can build a small UI that does the job like this :

function test(){
showURL("http://www.google.com")
}
//
function showURL(href){
  var app = UiApp.createApplication().setHeight(50).setWidth(200);
  app.setTitle("Show URL");
  var link = app.createAnchor('open ', href).setId("link");
  app.add(link);  
  var doc = SpreadsheetApp.getActive();
  doc.show(app);
  }

If you want to 'show' the URL, just change this line like this :

  var link = app.createAnchor(href, href).setId("link");

EDIT : link to a demo spreadsheet in read only because too many people keep writing unwanted things on it (just make a copy to use instead).

EDIT : UiApp was deprecated by Google on 11th Dec 2014, this method could break at any time and needs updating to use HTML service instead!

EDIT : below is an implementation using html service.

function testNew(){
  showAnchor('Stackoverflow','http://stackoverflow.com/questions/tagged/google-apps-script');
}

function showAnchor(name,url) {
  var html = '<html><body><a href="'+url+'" target="blank" onclick="google.script.host.close()">'+name+'</a></body></html>';
  var ui = HtmlService.createHtmlOutput(html)
  SpreadsheetApp.getUi().showModelessDialog(ui,"demo");
}

Solution 3:

There really isn't a need to create a custom click event as suggested in the bountied answer or to show the url as suggested in the accepted answer.

window.open(url)1 does open web pages automatically without user interaction, provided pop- up blockers are disabled(as is the case with Stephen's answer)

openUrl.html

<!DOCTYPE html>
<html>
  <head>
   <base target="_blank">
    <script>
     var url1 ='https://stackoverflow.com/a/54675103';
     var winRef = window.open(url1);
     winRef ? google.script.host.close() : window.alert('Allow popup to redirect you to '+url1) ;
     window.onload=function(){document.getElementById('url').href = url1;}
    </script>
  </head>
  <body>
    Kindly allow pop ups</br>
    Or <a id='url'>Click here </a>to continue!!!
  </body>
</html>

code.gs:

function modalUrl(){
  SpreadsheetApp.getUi()
   .showModalDialog(
     HtmlService.createHtmlOutputFromFile('openUrl').setHeight(50),
     'Opening StackOverflow'
   )
}    

Solution 4:

Google Apps Script will not open automatically web pages, but it could be used to display a message with links, buttons that the user could click on them to open the desired web pages or even to use the Window object and methods like addEventListener() to open URLs.

It's worth to note that UiApp is now deprecated. From Class UiApp - Google Apps Script - Google Developers

Deprecated. The UI service was deprecated on December 11, 2014. To create user interfaces, use the HTML service instead.

The example in the HTML Service linked page is pretty simple,

Code.gs

// Use this code for Google Docs, Forms, or new Sheets.
function onOpen() {
  SpreadsheetApp.getUi() // Or DocumentApp or FormApp.
      .createMenu('Dialog')
      .addItem('Open', 'openDialog')
      .addToUi();
}

function openDialog() {
  var html = HtmlService.createHtmlOutputFromFile('index')
      .setSandboxMode(HtmlService.SandboxMode.IFRAME);
  SpreadsheetApp.getUi() // Or DocumentApp or FormApp.
      .showModalDialog(html, 'Dialog title');
}

A customized version of index.html to show two hyperlinks

<a href='http://stackoverflow.com' target='_blank'>Stack Overflow</a>
<br/>
<a href='http://meta.stackoverflow.com/' target='_blank'>Meta Stack Overflow</a>

Solution 5:

Building of off an earlier example, I think there is a cleaner way of doing this. Create an index.html file in your project and using Stephen's code from above, just convert it into an HTML doc.

<!DOCTYPE html>
<html>
  <base target="_top">
  <script>
    function onSuccess(url) {
      var a = document.createElement("a"); 
      a.href = url;
      a.target = "_blank";
      window.close = function () {
        window.setTimeout(function() {
          google.script.host.close();
        }, 9);
      };
      if (document.createEvent) {
        var event = document.createEvent("MouseEvents");
        if (navigator.userAgent.toLowerCase().indexOf("firefox") > -1) {
          window.document.body.append(a);
        }                        
        event.initEvent("click", true, true); 
        a.dispatchEvent(event);
      } else {
        a.click();
      }
      close();
    }

    function onFailure(url) {
      var div = document.getElementById('failureContent');
      var link = '<a href="' + url + '" target="_blank">Process</a>';
      div.innerHtml = "Failure to open automatically: " + link;
    }

    google.script.run.withSuccessHandler(onSuccess).withFailureHandler(onFailure).getUrl();
  </script>
  <body>
    <div id="failureContent"></div>
  </body>
  <script>
    google.script.host.setHeight(40);
    google.script.host.setWidth(410);
  </script>
</html>

Then, in your Code.gs script, you can have something like the following,

function getUrl() {
  return 'http://whatever.com';
}

function openUrl() {
  var html = HtmlService.createHtmlOutputFromFile("index");
  html.setWidth(90).setHeight(1);
  var ui = SpreadsheetApp.getUi().showModalDialog(html, "Opening ..." );
}