Your system is trying to tell you that /data/steam/ubuntu12_32/steam-runtime/i386/lib/i386-linux-gnu/libgcc_s.so.1 is too old to be used with your version of mesa (/usr/lib32/xorg/modules/dri/r600_dri.so), making this issue report a duplicate of https://github.com/ValveSoftware/steam-runtime/issues/13.
Output of Steam:
(displays a window with that message, and then fails to open any game without crashing)
Workaround: https://bbs.archlinux.org/viewtopic.php?pid=1433884#p1433884
(still spews out a lot of errors, but at least allows games to run without crashing)
I believe that this may be an issue between how Ubuntu and Arch Linux name their libraries. It may also be an issue with the libraries which are packaged with steam. Xorg also recently updated to version 1.16, so, that might be it as well. This appears to be a problem with just ATI cards.
Contents of glxinfo:
Card:
01:00.0 VGA compatible controller: Advanced Micro Devices, Inc. [AMD/ATI] Juniper XT [Radeon HD 5770]