Exactly same thing on Fedora 19.

I installed Steam from http://spot.fedorapeople.org/steam/ and everything works perfectly, but when I launch "Big picture" SELinux blocks it with message

SELinux is preventing /home/Dāvis/.local/share/Steam/ubuntu12_32/steam from using the execheap access on a process.

*****  Plugin allow_execheap (53.1 confidence) suggests  *********************

If you do not think /home/Dāvis/.local/share/Steam/ubuntu12_32/steam should need to map heap memory that is both writable and executable.
Then you need to report a bug. This is a potentially dangerous access.
Do
contact your security administrator and report this issue.

*****  Plugin catchall_boolean (42.6 confidence) suggests  *******************

If you want to allow selinuxuser to execheap
Then you must tell SELinux about this by enabling the 'selinuxuser_execheap' boolean.
You can read 'None' man page for more details.
Do
setsebool -P selinuxuser_execheap 1

*****  Plugin catchall (5.76 confidence) suggests  ***************************

If you believe that steam should be allowed execheap access on processes labeled unconfined_t by default.
Then you should report this as a bug.
You can generate a local policy module to allow this access.
Do
allow this access for now by executing:
# grep steam /var/log/audit/audit.log | audit2allow -M mypol
# semodule -i mypol.pp

Additional Information:
Source Context                unconfined_u:unconfined_r:unconfined_t:s0-s0:c0.c1
                              023
Target Context                unconfined_u:unconfined_r:unconfined_t:s0-s0:c0.c1
                              023
Target Objects                 [ process ]
Source                        steam
Source Path                   /home/Dāvis/.local/share/Steam/ubuntu12_32/steam
Port                          <Unknown>
Host                          Fedora
Source RPM Packages           
Target RPM Packages           
Policy RPM                    selinux-policy-3.12.1-71.fc19.noarch
Selinux Enabled               True
Policy Type                   targeted
Enforcing Mode                Enforcing
Host Name                     Fedora
Platform                      Linux Fedora 3.10.7-200.fc19.x86_64 #1 SMP Thu Aug
                              15 23:19:45 UTC 2013 x86_64 x86_64
Alert Count                   1
First Seen                    2013-08-22 21:36:30 EEST
Last Seen                     2013-08-22 21:36:30 EEST
Local ID                      fe053ad2-a729-4f22-aa23-aef08e8fe32c

Raw Audit Messages
type=AVC msg=audit(1377196590.821:485): avc:  denied  { execheap } for  pid=3960 comm="steam" scontext=unconfined_u:unconfined_r:unconfined_t:s0-s0:c0.c1023 tcontext=unconfined_u:unconfined_r:unconfined_t:s0-s0:c0.c1023 tclass=process


type=SYSCALL msg=audit(1377196590.821:485): arch=i386 syscall=capget success=no exit=EACCES a0=c349000 a1=c000 a2=7 a3=ff89291c items=0 ppid=3328 pid=3960 auid=1001 uid=1001 gid=1001 euid=1001 suid=1001 fsuid=1001 egid=1001 sgid=1001 fsgid=1001 ses=1 tty=(none) comm=steam exe=2F686F6D652F44C4817669732F2E6C6F63616C2F73686172652F537465616D2F7562756E747531325F33322F737465616D subj=unconfined_u:unconfined_r:unconfined_t:s0-s0:c0.c1023 key=(null)

Hash: steam,unconfined_t,unconfined_t,process,execheap

then in terminal I just enable it with

$ setsebool -P selinuxuser_execheap 1

and next time I launch steam it works in "Big Picture" mode good :)
just need document this step somewhere I think.