Yes and no.
I have a 680 lying around which is near identical to the 770 and it runs doom at 60fps on low.
That's one of the less demanding triple A titles too.
If you're at 1080p and don't mind lower quality settings, yes it can manage current games, but i'd upgrade to something like the MSI Gaming X RX 480 which is currently an absolute bargain.
Considering that card will last another year at most, i'd upgrade now and get it out of the way.

It'll be well worth it in the long run.
PCPartPicker part list /
Price breakdown by merchant
Video Card: MSI Radeon RX 480 4GB GAMING X Video Card ($179.99 @ Newegg)
Total: $179.99
Prices include shipping, taxes, and discounts when available...