In your top photo you can see where the cable-end attaches to the top of the pedal. How is the cable attached? Is it easy to disconnect right there? I looked in the shop manual & can't find anything on the accelerator pedal itself.

I guess if you can disconnect the cable then you can push/pull on the pedal and the cable separately. That's how to tell which one is sticking.
If it's the pedal, you can clean & grease that pivot pin & spring that is visible in your top photo.

If it's inside the cable, then I think cable replacement is the way to go. I don't know how you'd tell the difference between lubrication turning to gunk vs. a kink somewhere in the cable. And I bet it's a real pain to work some grease through the cable...

 

On your bottom picture, the cable in back is the cable attached to the gas pedal. The accelerator cable looks tight, but it is hard to tell from the picture.. That cable should have slight slack with the gas pedal released. See the linked video below where he shows the proper amount of slack.

You can also try spraying some wd-40 (or equivalent) on the cable where the cable meets the rubber boot in your picture, then press and pull the gas pedal from inside the car to work in the wd-40.

 

The WD-40 worked! You learn something new everyday. Thanks to you all for your help once again.

 

I've been away from my computer for a little while...

It's great that it loosened up.
But I think that the WD-40 might dry out after awhile, & put you back where this started. Maybe you can work some better oil into the cable? A little bit of engine oil squirted in from the engine compartment?
(WD-40 is great for getting things like this to loosen up. But not so good as a long-term lubricant)

 

Hey Jim, so that turned out to be right. I'm having to put WD 40 on it every couple of weeks. What kind of oil would you suggest? I've got a bunch of different kinds laying around the house.

 

Almost anything. I always seem to have some engine oil laying around that's different weight from cars I don't own anymore. I use that for things like door & hood hinges where the exact oil specs don't really matter.

 

Perfect I'll try that!