Will a throttle cable stretch over time?

Yes, they can stretch a little over time and strain. But they can only stretch a little as they are metal which can only stretch so much.

There are two basic components in a cable. You have the conduit (the outer housing/tube) and the strand (the part that goes into the tube).

If you want to try to "tighten" it back up, you may try to crimp a fishing weight (the small round type) to the end where the pedal goes, just under the original "button" end that the throttle works off of. (You can't crimp it at the carb end, as it will interfere with the hook up to the carb). Maybe try to crimp the old round crimp on metal fishing weights on the end just before the "button) to essentially make the strand "shorter" as the car sees it. Be careful doing this as it may interfere with installing the strand into the pedal end. Maybe crimp the weight on while the cable is still installed.

The best way to fix this is to replace the old worn out cable assembly with a new one.