Absolutely you can do this. Many companies all over the world run a portion (sometimes all) of their commercial infrastructure on Linux distributions. Ubuntu in this sense is no different from Red Hat, Debian, SUSE, and the rest.
Just because the kernel and many system components are under the GPL, that in no way means the code you run on top of it must also be free or open source software. A simple search for commercial software and commercial web services that run on Linux will prove this quickly.
One simple example would be Google. Virtually their entire server infrastructure runs on Linux, but clearly that hasn't forced them to give out the code to their web indexing software, or Gmail, or G+ (not to mention the entire Android platform which runs on Linux).
As for being sure enough to convince your legal department that it is ok to run Ubuntu Server, you might consider buying an Ubuntu Advantage support contract from Canonical. They (ok, we) include legal indemnification with each support contract. Please see:
Also: I didn't mention EC2 specifically because there is no reason to think that hosting it on EC2 would be any different compared with running it "in house" from a legal standpoint.
Also, too: In the case of Android, let me clarify. Android's modifications to the Linux kernel are GPL'd, however the rest of the platform is on a very permissive Apache-style license, and obviously a great many companies sell paid, closed-source software that runs on it.