Have you tried the workaround?
sudo cp /usr/share/applications/google-chrome.desktop ~/google-chrome.desktop.bak~
sudo sed -i 's/\/usr\/bin\/google\-chrome\-stable/\/usr\/bin\/google\-chrome\-stable \-\-touch\-devices=123/g' /usr/share/applications/google-chrome.desktop
If this workaround causes problems, use this command to uninstall:
sudo cp ~/google-chrome.desktop.bak~ /usr/share/applications/google-chrome.desktop
Source: https://code.google.com/p/chromium/issues/detail?id=456222#c175
Following @muru's good suggestion, in a comment to my question, I have filed a bug report in Launchpad, regarding this issue:
Bug #1547762 “SSL Protocol Error” in Chromium for several Google web sites after installing Ubuntu Security Updates for libnss3 : Bugs : chromium-browser package : Ubuntu
https://bugs.launchpad.net/ubuntu/+source/chromium-browser/+bug/1547762
That bug report has now the information that "This bug affects you and 12 other people". The Status of the bug was changed from 'New' "to 'Confirmed' because the bug affects multiple users".
So, it seems this is indeed a bug. I'll keep updating this answer according to the progress of that Bug Report.
NEWS FLASH (Feb 25, 2016): This issue was apparently fixed by a Software Update to Chromium that was made available today! For more detail, read "Update #3 (Feb 25, 2016)" further down:
Update #1 (Feb 22, 2016):
Based on the bug template, you've indicated you're running version
"37.0.2062.120", which is no longer supported, and has a number of
critical security bugs. You should upgrade to the latest version of
Chrome, which is Chrome 48.
If you're running a version provided by another party, such as your
distro, please inform them that they're distributing an insecure
version.
You can download the latest stable version of Chrome at
https://www.google.com/chrome/
Bug #1468666 “Chromium VERSION on Ubuntu 12.04 LTS” : Bugs : chromium-browser package : Ubuntu
https://bugs.launchpad.net/ubuntu/+source/chromium-browser/+bug/1468666
Update #2 (Feb 23, 2016):
The bug that I had opened ("Bug #1547762 "SSL Protocol Error" in Chromium for several Google web sites after installing Ubuntu Security Updates for libnss3") was marked as being a duplicate of the following bug, reported on November 27, 2015 (affecting now 53 people, still marked as "Unassigned "):
Bug #1520568 "All queries fails when 'google' is used: ERR_SSL_PROTOCOL_ERROR" : Bugs : chromium-browser package : Ubuntu
https://bugs.launchpad.net/ubuntu/+source/chromium-browser/+bug/1520568
Update #3 (Feb 25, 2016):
This issue seems to be fixed by Software Updates to Chromium that were made available today! :-) For the Bug 1520568 "All queries fails when 'google' is used: ERR_SSL_PROTOCOL_ERROR", the user "Marc Deslauriers (mdeslaur)" and the bot "Launchpad Janitor (janitor)" added information regarding a patch made available today and reported that "This bug was fixed in the package chromium-browser - 37.0.2062.120-0ubuntu0.12.04.2". The Status of that Bug got consequently changed from "Confirmed" to "Fix Released".
So, in "Comment #47" of that page, the following info appears:
Launchpad Janitor (janitor) wrote 21 hours ago: #47
This bug was fixed in the package chromium-browser - 37.0.2062.120-0ubuntu0.12.04.2
chromium-browser (37.0.2062.120-0ubuntu0.12.04.2) precise-security; >urgency=medium
- debian/patches/nss-321-fix.patch: fix compatibility with nss 3.21.
(LP: #1520568)
-- Marc Deslauriers Wed, 24 Feb 2016 13:42:57 -0500
And, in fact, the "Update Manager" was "filled" with several available updates, including the 3 patches below for Chromium. I tested closing "Chromium" and then selecting only those 3 patches, installed them and then started Chromium again, to see if those patches were enough to solve the problem... and they were enough! :-)
So now, Chromium is now again able to open Google web sites. Obviously, I suggest that you also install the other available patches/Security updates. For now, I installed only the 3 available patches for chromium, because I wanted to see if those were enough to solve the issue.
Under these circumstances, I believe that I can now mark this question as answered :-)
Update information for Chromium packages:
Chromium browser
chromium-browser (Size: 45,4 MB)
Changes for the versions:
Installed version: 37.0.2062.120-0ubuntu0.12.04.1~pkg917
Available version: 37.0.2062.120-0ubuntu0.12.04.2
Version 37.0.2062.120-0ubuntu0.12.04.2:
- debian/patches/nss-321-fix.patch: fix compatibility with nss 3.21.
(LP: #1520568)
chromium-browser language packages
chromium-browser-l10n (Size: 2,9 MB)
Extra ffmpeg codecs for the Chromium Browser
chromium-codecs-ffmpeg-extra (Size: 831 kB)
Best Answer
From the staging PPA for Chromium, it seems the builds are failing for some reason. The packagers maybe aware of it, a bug has been filed. You can mark it as affecting you.