I also have this problem now, but I'm sure I did NOT have it about month ago.
To be more precise, ping apps on S4 (that I tried) are not returning verbose errors, but they all result in ping timeouts if I ping from S4 to anything, including S4 local IP address.
On the other hand, I can ping from PCs in local network to phone, I can access internet from phone, I can access local LAN services (SMB sharing, video streams, RDP...) from phone. Only thing I can NOT do is use any app on phone to ping from phone to anything. It is not like S4 is blocking ICMP (since ping from PC to S4 works), it is more like S4 is preventing ping API to work correctly.
While my phone is not rooted, it was not rooted month ago, and back then it was able to ping. Maybe it is some of recent Samsung patches received OTA?
I'm on I9505XXUDMH6 currently (3.4.0-1220396), international S4 with Snapdragon 600.