What is your OS? Windows 7 64 Bit?
Do you have at least 8Gb of Ram?
The 7770 is a little worse than the 660 which is the recommended BUT it will suffice indeed, if you are considering an upgrade anytime soon I suggest getting the 7850 or 7870 the 7870 could play BF3 on Ultra so it might do a good job of doing close to the same in BF4.
Or you could just wait a bit longer new stuff will be out soon... you want a GPU with at least 3GB of memory though so that 1GB is holding you back.
When did you get that pc? If it was 3 years ago or so I would suggest considering a new 1 or at least an upgrade (by upgrade I mean keep the box/chassis, PSU, Harddrives, keyboard, mouse, headset, monitor and even the Ram) replace the hardware inside the chassis.
A nice Sabertooth 990FX
Radeon 7870
AMD FX6300 (8350 if budget allows)
(old ram as long as it was 8GB of 1600Mhz, your Mhz is right but I am not sure if you have 4GB or 8GB)
You might need to do a clean install of the OS since it's a new motherboard...
That upgrade should last you another 3 years, then after the 3 years you might just need 4GB of Ram and a new graphics card (which is really not that expensive.)
At least Xbox 360 runs Battlefield 4 man

I am surprised it can even run it to be honest.
Potato>Xbox 360
Toaster with Blu Ray drive> PS3