No I don't think so. Presuming that we're talking about a cut,
healed over to me just indicates that a scab has formed, closing the wound - but not necessarily that the skin has yet regrown.
Healed up on the other hand means just that - that the wound is completely closed. It's another of the examples of up being used to give a sense of "completion", as in
Eat up your spinach or
Tidy up this room.
However bear in mind that it's actually more usual to use the verb heal without any particle at all. Entering the phrases into a
concordancer, which searches 56 million words of a corpus containing both British and American English, I got a full page of examples with
heal (ie 40 - but that's the limit of the number of examples shown), but only five with
heal up and none at all for
heal over.
Here are some of the examples which came up for for the first two :
... established burns are slow to heal and can easily become infected
... facilitates drainage until damaged veins can heal and ...
... a natural antiseptic, to help heal and soothe warts quickly.
The physical scars heal but sometimes the mental scars never do.
... As skin ages, its ability to heal and repair itself ...
If nappy rash does occur, and doesn't heal up quickly or is making the baby unhappy, ...
Some sores are more severe and take longer to heal up.
... expect the holes in her nose will heal up with time," said Anthea.