Can Kentucky Derby winners run in the Melbourne Cup?

Yes. The race is a "quality handicap" open to horses three years old and older, who qualify based on earnings and performance. So, theoretically, a Kentucky Derby winner could run in it, but I do not know if any ever have.

https://en.wikipedia.org/wiki/Melbourne_Cup

The distances of the races are quite different, and it is unlikely that a horse bred to run in the Kentucky Derby (2km, and no weight handicaps) would be suitably bred to be successful in the Melbourne Cup (3.2 km handicap, with penalty weights for major wins). Since it is quite an expensive proposition to get the horse there, and keep it in quarantine for the required time, it is unlikely that it will happen.

https://en.wikipedia.org/wiki/Kentucky_Derby

https://en.wikipedia.org/wiki/Melbourne_Cup

Although the distances and WFA vs handicap arrangements are different for the two races, Kingston Town was a champion three-year old who won the AJC Derby (the aussie equivalent to the Kentucky Derby) and went on to race twice in the Melbourne Cup (unplaced and second). A winner of the Kentucky Derby can race in the Melbourne Cup, if they gather enough wins/points in the appropriate races to qualify; and make the 24-runners cut-off (the aussie tote machines can't handle bigger fields).

http://www.horsedirectory.com.au/racinghistory/horse_profiles/hpro.KingstonT.html

Secretariat won the Kentucky Derby (and then went on to win the Triple Crown). We never saw Secretariat race in Australia in the flesh, but one of his progeny did. Kingtson Rule won the Melbourne Cup in 1990, and set a new record. The record still stands today.

http://www.horseracegame.com/community/content/blogs/stretchrun/02-02-2010/secretariat-didnt-reproduce-himself-sired-many-champions-and-was