Problem Downloading the City Nav Europe from Garmin Site

Are you saying that even when the Europe map is downloaded onto the SD card there is still no room for it? That of course does not make sense.
What you should do is this: make sure that the download is indeed directed at the SD card (it is presented by your PC as a different "disk drive" than the Nuvi's internal memeory). At the end of the download the SD card should contain a root directory named \Map, which will contain a *.img file, a *.unl file, and a *.gma file. The latter being the file needed for map authentication.
By the way it would be a good idea to back all files up, particualrly those on the Nuvi's internal memory. That way you can even erase the US map at will, then put it back later.

There is plenty of room on the Micro SD. It did download the three files in a folder labeled \map. The problem is that it does not recognize the Europe map set. After the first failed download I called garmin. they told me to delete the \map folder and re install. They felt there was a problem with the installation because it took over 48 hours to download. I deleted \map and am reinstalling now. I am at 18 hours and am at 80% installation complete.

on my 680 if you want one map on the sd card...

gmapsupp.img is placed in the folder /garmin/
6xx series only supports SD cards up to 4GB. SDHC will not work properly. The 3760 you could try the same garmin folder, see if it makes any difference.

The 1xxx, 3xxx models have a different folder and file structure than the older models such as the 680

I'm not sure about 3xxx models but 1xxx has the same folder structure as above mentioned: /Garmin/gmapsupp.img

Of course there is a \Garmin folder on the 1xxx, but it does not hold maps. Thus your /Garmin/gmapsupp.img
wouldn't appear on the unit's map-list menu. on 1xxx maps must be stored either in the hidden folder .system or in \map

But only for the maps stored on internal memory. In this case he is asking about maps stored on a SD card. His mistake is that the root directory shouldn't be /map/gmapsupp.img but /garmin/gmapsupp.img.

On 1xxx and 3xxx, maps must be stored inside the \map root directory - either on internal memory or SD !!! And by the way, in that directory they may be named 'anyname.img', not just gmapsupp

and that is not correct for this particular model. When on internal memory all supplemental maps are stored in directory /map/anyname.img, and you can have more than one map in it. When loaded to a SD card map has to be named gmapsupp.img and stored in /garmin/ directory.

Well, if you do own that model and speak from experience - then I stand corrected However for most other 13xx and 14xx models it is not so

Over night my second download attempt has the same results as the first. The garmin site gave me a download complete message. when I turned on the unit (Nuvi 3760 LMT)a message came up that said "cannot authenticate maps). when I go to the tools and look at "My Maps", the only one visible is the North America city navigator (which still works fine). When I plug the Garmin back into the computer and look at the files on the 16GB Micro SD plugged into the Nuvi, here is what I see:

Removable Disk (P:)
Folder "maps"
In the maps folder, three files (and their size):
Europe_2012_10.gma (2.53KB)
Europe_2012_10.img (2.16GB)
Europe_2012_10.unl (25 byes)

It seems to me that I have the right files but the device isn't seeing them, but that is just a guess on my part.

The first thing that caught my eye was the folder name. It should be map and not maps (i.e. singular, not plural); at least that's the way it is on my 1490 - are you sure yours is 'maps' or is this just a typo? If it is maps indeed, change it to "map" and try again.
BTW it looks as if you do have a complete download, so back it up to avoid having to download again.

Update: however I suspect that your unit may also suffer from this disease:
http://www.poi-factory.com/node/31376
i.e. make also sure that it runs the most current firmware.