Does my 2010 MacBook Pro 13 have a bad GPU

Your system is OK!

13” MacBook Pro 2.4 GHz Core 2 Duo (Mid-2010)

Your system only offers an intergraded graphics chip which is a combo NVIDIA GeForce 320M. Unlike other Intel CPU’s this one does not offer its own graphics engine Intel Core 2 Duo Processor P8600 which can confuse people. Here’s an example of what your system looks like architecturally:

So clearly the internal display is working so your GPU is OK.

What is more likely the settings of the external display are not properly set. So lets walk through them, from EveryMac:

Apple reports that this system also supports "1152 by 720, 1024 by 640, and 800 by 500 pixels at 16:10 aspect ratio; 1024 by 768, 800 by 600, and 640 by 480 pixels at 4:3 aspect ratio; 1024 by 768, 800 by 600, and 640 by 480 pixels at 4:3 aspect ratio stretched; 720 by 480 pixels at 3:2 aspect ratio; 720 by 480 pixels at 3:2 aspect ratio stretched."

So you’ll need to manually set the display to one of these choices. That should fix things.