How do I open a web browser (URL) from my Flutter code?

TL;DR

This is now implemented as Plugin

const url = "https://flutter.io";
if (await canLaunchUrl(url))
  await launchUrl(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.