Is there a genetic link between North America and Vikings?

The term "blue eyes" is a misnomer. This refers to the appearance of light eyes in the tribes: a dark grey or green, most prevalent among the Mandans. It's re-appearance in some of the Snowbird Cherokee, is most likely due to later inter-breeding (aka: after the Trail of Tears). However, in the Algonquin legends it is well supported that the Norse did, indeed, interbreed with the Native groups. There existed no dissuading considerations for this. The Norse produced larger and hardier offspring. Offspring that were always viewed as members of their tribe and clan. Hardier offspring were considered a boon. Higher reproduction rates were a distinct advantage. Merely the broader hip span produced by the introduction of Western European lineage into the tribes, insured more successful births to the generations which followed this initial cross-fertilization.

Nors*m*n traded with the Thule of the Arctic region, whom they called Scraelings. The Thule (Proto-Inuits) arrived in the northeast Atlantic zone not long after the Norse settled in Greenland. There was a paleolithic, pre-Thule population in the Atlantic region, called the Dorset culture. They were annihilated by the Thule. The Thule also prevented the Norse from permanently settling any further West than eastern Greenland. Ultimately, they drove the norse entirely out of Greenland, so there were no Nors*m*n in the Americas. (Greenland is considered to be North America.) This left the Thule as the sold inhabitants of the Arctic zone.

The Thule came from the Bering Strait. The region was connected with Asian trade, and utilized iron. There was a lot of competition and warfare based around hunting grounds for the bow-headed whale. They had Mongolian bows and Chinese style slat armor, which they made from bone. This made them militarily superior to the Norse. Recent research suggests that Thule migrated from the Bering Strait to the Atlantic because Genghis Khan disrupted the iron trade from Asia. Searching for new sources, they crossed to the continent with dog sleds in under 5 years.

Nors*m*n couldn't have taken Thule women as they pleased. I think that such interaction would probably have been in the form of kidnapping or trade, so the natives would be assimilated into the Norse. If they did have sexual interactions with the Dorset people, it wouldn't have mattered because they were decimated. There were other types of natives in Newfoundland and Labrador, and a few Nors*m*n reached here. Given the paucity of Norse archaeological sites, or archaeological evidence thereof, its hard to imagine any significant interactions with anyone outside of Greenland.

Well, we never did get a reference for what "blue-eyed Cherokee"s you were actually talking about. However, the very first link I got on a google search happens to be a nice Native American urban myth debunking page that I've directed people to before, so I'll quote the appropriate passage from it for you:

Q: I heard that there was a tribe called the "blue-eyed Indians" because Norse or Celtic explorers intermarried with them. Is that true?

A: No. There is no tribe of Indians that is predominantly blue-eyed. In fact, blue eyes, like blond hair, is genetically recessive, so if a full-blood Indian and a blue-eyed Caucasian person had a baby, it would be genetically impossible for that baby to have blue eyes. Blue eyes only occur in people who have blue-eyed Caucasian relatives on both sides of their family tree, and even then only some of the time. There are tribes who have had plenty of blue-eyed individuals after colonization, such as the Lumbees and the Cherokees, because those tribes lived in close contact with a Caucasian community as large as their own and intermarried with them frequently. Before colonization, not a chance. A few Norse or Celtic explorers couldn't have left behind blue-eyed Indian babies any more than a few Caucasians exploring Africa could have left behind a race of blond-haired black people.

For the Cherokee in fact probably their most historic Chief, John Ross, who led them through the Trail of Tears, was blue-eyed (and possibly red-haired). How did this happen you might ask? Well his father and his maternal Grandmother were both Scottish. Native Americans prior to cultural assimilation didn't have the European concept of "blood". Instead they were clan-based (likely why they got on well with the Scots), which was more of a voluntary association. Sort of like how Europeans are with their football teams.

I am going to say Yes, to the title question, and No to the question in the body.

Is there a link...

There is a genetic link, but not in the direction you were expecting. An article published on the National Geographic website makes some claims concerning Native American DNA signatures located among Icelandic populations:

Analyzing a type of DNA passed only from mother to child, scientists found more than 80 living Icelanders with a genetic variation similar to one found mostly in Native Americans.

and

"We know that Vikings sailed to the Americas," said Agnar Helgason of deCODE Genetics and the University of Iceland, who co-wrote the study with his student Sigrídur Ebenesersdóttir and colleagues. "So all you have to do is assume … that they met some people and ended up taking at least one female back with them.

but

Despite the evidence, for now it's nearly impossible to prove a direct, thousand-year-old genetic link between Native Americans and Icelanders.

For starters, no living Native American group carries the exact genetic variation found in the Icelandic families.

In light of another answer mentioning Greenland, the DNA study looked at that as well:

The Inuit, often called Eskimos, carry no version of the variant—a crucial detail, given that Greenland has a native Inuit population

So this DNA sample cannot be traced to any interaction with the Inuit.

.

This leads us to the more specific query implied in the body of the question, essentially :
Were there blue-eyed Cherokee Native Americans descended from Vikings?

One problem with this is that technically we don't even seem to know what Cherokee DNA looks like to start wth. From a genealogy site, AccessGenealogy:

First, the readers should understand that if any commercial DNA lab returns tests results that state a percentage of DNA for a particular Southeastern Native American tribe, the report should be considered fraudulent. The American Society of Human Genetics has not certified any DNA test markers to be associated with a particular Southeastern American Indian tribe.

We don't have good DNA samples of any original,pre-colonial era North American Native American group so that we can isolate specific markers of that group. So to isolate Viking DNA parts in modern DNA, it would be very difficult to prove they were from original contact (abt. the year 1000) or later European contact during or after the colonial period.

Another website provides more discussion of the 'blue-eyed' issue here:Native Languages.org