Press Alt+F2, enter google-chrome --app=http://gmail.com then pin the icon (right click it then click 'Pin to Dock') for the window that appears.
chromium-browser --app=http://gmail.com is the equivalent command for Chromium.
This will work for other websites too.
In a Linux distribution, software doesn't usually use its own internal "automatic update" mechanism. Instead, it relies on the distribution to update it.
This is the case for both Firefox and Chromium supplied with Ubuntu.
The interesting thing here is that Firefox and Chromium have a different level of support from Canonical, and also that Firefox on Ubuntu has a somewhat unusual release strategy.
Firefox is included in Ubuntu's main repository, whereas Chromium is in the universe repository. This means that Ubuntu's version of Firefox receives the best level of support directly from Canonical, whereas Chromium's support relies on the Ubuntu community. Most packages in universe rely heavily on the work of Debian developers, and Debian has different priorities. So, based on this you would expect Firefox to include more Ubuntu-specific customisation, and to be better supported with updates.
The thing is, almost all software in a stable-release distribution (like Ubuntu, and most other distributions other that the "rolling release" ones) should not receive updates to new versions of software during the life cycle of that version of the distribution. At best, it should receive only minor updates or security patches. Firefox on Ubuntu is somewhat unusual here, because Ubuntu are providing full major new upstream versions of Firefox to their stable distributions.
The justification for this will be Firefox's rapid release strategy, which ensures that "major" releases are actually somewhat less disruptive than most software's major releases, but also makes it quite hard to backport security patches to older versions. So Canonical has obviously decided just to give Ubuntu the new upstream version than patch the old version.
For Chromium, it's not completely like this, but looks can be deceiving. Chromium on Ubuntu sometimes does get new upstream releases (which is why even Lucid is on a v18.x), and other times the Ubuntu community patch the existing version with fixes and security updates, but don't increment its version number. However, this means that despite the version number of Chromium in Ubuntu, it still includes some of the fixes that newer versions of Chromium have.
Best Answer
You are running Ubuntu 14.04 as to be seen from the 'tag' in your question. This system is about three years old and the repositories provide the chromium-browser 53.0.2785.143 package. So, you should consider to install Ubuntu 16.04 LTS, it provides the chromium-browser 55.0.2883.87 package, which most probably will get updated for some longer time - at least until the next LTS edition gets released.