An attempt at fixing this has been made in the latest beta version of the launcher. Please try that before trying any workarounds. You can toggle the beta version on in the Settings pane - If you are stuck on the login screen, try the following:
- Locate your .minecraft folder. In that folder, open the 'launcher_profiles.json' using notepad. At the bottom of the file change "channel": "release", to "channel": "beta"
Please post a new launcher_log.txt if you still have this problem in the beta.
Some users have reported that this issue can be worked around by changing the DNS. (Guide for switching to Google's DNS by Ethan Paulakonis in this comment)