TL;DR

This is now implemented as Plugin

const url = "https://flutter.io";
if (await canLaunch(url))
  await launch(url);
else 
  // can't launch url, there is some error
  throw "Could not launch $url";

Full example:

import 'package:flutter/material.dart';
import 'package:url_launcher/url_launcher.dart';

void main() {
  runApp(new Scaffold(
    body: new Center(
      child: new RaisedButton(
        onPressed: _launchURL,
        child: new Text('Show Flutter homepage'),
      ),
    ),
  ));
}

_launchURL() async {
  const url = 'https://flutter.io';
  if (await canLaunch(url)) {
    await launch(url);
  } else {
    throw 'Could not launch $url';
  }
}

In pubspec.yaml

dependencies:
  url_launcher: ^5.7.10

Special Characters:

If the url value contains spaces or other values that are now allowed in URLs, use

Uri.encodeFull(urlString) or Uri.encodeComponent(urlString) and pass the resulting value instead.

If you target sdk 30 or above canLaunch will return false by default due to package visibility changes: https://developer.android.com/training/basics/intents/package-visibility

in the androidManifest.xml you'll need to add the following directly under <manifest>:

<queries>
    <intent>
        <action android:name="android.intent.action.VIEW" />
        <category android:name="android.intent.category.BROWSABLE" />
        <data android:scheme="https" />
    </intent>
</queries>

Then the following should word

const url = "https://flutter.io";
if (await canLaunch(url)){
    await launch(url);
} else {
    // can't launch url
}

For Flutter:

As described above by Günter Zöchbauer

For Flutter Web:

import 'dart:html' as html;

Then use:

html.window.open(url, name);

Make sure that you run flutter clean if the import doesn't resolve.