This will typically happen when OS X's firewall is active. Try going into System Preferences -> Security -> Firewall and disabling your firewall. It can also happen if a specific address is blocked from receiving certain ports. Your router should be set to forward/allow the following ports to your machines:
Blizzard Downloader (TCP) 1119, 1120, 3724, 4000, 6112, 6113, 6114, 6881-6999 (UDP) 1119, 1120, 3724, 4000, 6112, 6113, 6114, and 6881-6999
Diablo III (TCP) 80, 1119, and 6881-6999 (UDP) 1119, 6881-6999
World of Warcraft Launcher and Game Client (TCP) 3724, 1119, 1120, 4000, 6112, 6113, 6114, 80, and 6881-6999 3724, (UDP) 1119, 1120, 4000, 6112, 6113, 6114, 80, and 6881-6999
The following support article details the ports in a more visually friendly manner:https://us.battle.net/support/en/article/firewall-configuration-for-blizzard-games
The World of Warcraft Launcher is listed because Diablo III uses an identical Launcher and requires the same ports open.
If your router has the capability, you should try using the DMZ function specifically for your Mac's IP. This will open all
ports and allow you to install. Be aware though that this will also leave the machine open to malicious requests, which while they cannot do anything on their own, can in conjunction with malware cause problems. The DMZ option should be considered a last resort, even on OS X.
Could you give me the make and model of both your cable/DSL modem and your router (if you are using separate components)? It might yield an avenue not yet considered here.