No it isn't a legit fear. Cross-pollination is only an issue if you are saving seeds from the fruit to grow the following year. CP affects only the seeds, not the fruit itself. Otherwise we'd all be eating some really weird stuff. :)

Some vegetables will easily cross, some the odds are quite low. So while next year those saved seeds, if you saved them for some reason, might grow you a cumbermelon but not this year. If saving seeds then isolation of varieties and steps like bagging blooms and doing hand pollination are required.

Dave

That's good information. The fruit is like the womb of the plant.....it carries the seed and the seed is the result of any cross pollination. The fruit will be consistent with the parent plant.

Dave covered pretty much all the bases.
Another point is that not everything within cucurbitacea can be crossed. For example, you cannot cross cucumber and watermelon, you cannot cross melon with watermelon..so forth.

Obviously your coworker has never had a garden.

Or is well supplied by erroneous garden lore. (A common affliction!)