Can I clear this up a little?
Instead of nations the 2 beligerants of D-day were the Americans and "Commonwealth Nations and its alliances".
The commonwealth nations consisted of Canada, The United Kingdom of Great Britain and Northern Ireland, Indian troops, Free French, Polish, Czech and a few other nations. Quite a lot were involved with the Airforce and Navy rather than ground troops.
French Commandos played key roles in the D-day landings around Port En Bessin and really should not be ignored at all as it supported the link up between the Commonwealth landings and the American landings.
The Commonwealth had ups and downs for its success, Pegasus Bridge was taken in 8 minutes and Horsa followed up shortly afterwards (You would not beleive how close the Gliders landed, you can pace it out for about 40 steps, its not even a light jog) ... to a horrific assault on the Merville Battery of 50% casualties from a force that was already depleted to 25% from Ad-Hoc parachute drops ... which is quite scary.
If you think that Omaha and Utah beach were horrible? well the total losses were around 10% of the forces deployed to take the beaches ... they expected 60% losses ... its not really compareable. You can find the Current Cemetary on Google maps ... just imagine it 6 times bigger.
While some people are now saying "Europe would be beaten" well yes your right there, but America would of soon fallen, U-boats already at its doors only 10 miles from its shores, the 2 Biggest battleships ever built would still be at large. America would of been sandwiched between 2 nations, its ports on the east bloodied at the nose ... and to their west Pearl harbour funneled into a tiny gap.
America is fortunate never to have a war on its doorstep thus far, while most other nations have already sorted out defence measures and already have experiance in thus matters. America always suffered the most due to lack of experiance in tactical warefare, it is why its losses were so high in WWI, the same for WWII ... it joined in late and suffered for it in its soldiers and overzealous leaders. The Americans Criticised the British and canadians for being "Slow", well too right in my eyes, they had been fighting far longer, knew how to win their battles without sustained losses ... and this is where the overzealousness comes in, the losses on Omaha were so severe it dwarfed the combined effort of the 4 other beaches, it was not the hardest beach they were all relatively Equal.
The Use of DD Tanks were a key role, on the other beaches the commanders had opted to land the Tanks rather than Float them, but on Omaha well you can see the results. Of the near 50 tanks deployed only 2 made it to shore ... where other Beaches had had near all of their tanks land successfuly with landing them conventionaly. Would Omaha of been so bad with tanks on the Field? ... I think the losses would of been significantly less.
If you think America would of been fine without joining in at all, you would be sorely mistaken, what happened to the countries that Germany conquered? Well they joined it, there are various accounts of Polish and Czech regiments, even Russian regiments in the German army who were conscripted AFTER their conquest. Could America Survive a dual attack from 2 nations that have far more Experiance than it of Aqatic assaults, Naval Sieges and Experiance by the bucket load of Land experiance.
Michael Wittmann is a prime example, with over 100 russian tank kills before moving over the western front and totalling a British Armoured Collumn with his Comrades on his very first mission of the western front ... you think America would of been safe from him with its minimal Tank fighting experiance?
Would America of Survived without joining in the war? Doubtful - its lack of experiance and willingness to have U-boats so close to its borders pre-war point to sloppy mistakes which lead to thousands of deaths.
I would suggest going on a tour of the Battlefields you'll learn a lot about these "What if" Scenarios of WWII, after going around Normandy you can really judge how horrible and horrific it must of been for those people lost, not just on the Allies but Axis too ... the German Cemetary in La Cambe has 26,000 burried in it ... the American Cemetary has .... 26,000.
Quite an eye opener really when you go there and I would heavily suggest it to learn something and see what its like. Pay respects to both as Equals, neither side was better than the other, neither was braver, neither was more cowardly ... No matter how you look at it ... Thousands of people lost their lives ... and you probably knew none of them or what they were like ... but there they are all gathered in one place, its scary to imagine each stone as one person (Or in the German cemetary 2-5 people).