We'd like to help....Problem is a detailed knowledge of Danish social and religious history would be needed to fully and accurately answer your question. You might have a better chance at such an answer in the Denmark General forum.
This website claims that in 1850 eleven percent of births in Denmark occured outside of marriage:http://www.understandingyourancestors.com/ar/parishBirth.asp...
I'll try to help based on my reading in Danish history (limited), study of Danish genealogy boards, and research of Danish records. I'm not an expert.
> "If a single woman became pregnant, did they leave their families because of shame? Was the stigma unbearable that a girl/woman would run away in the 1860s to the USA." < I'm not familiar with a case where a girl left her family out of shame. To the contrary I have seen many many Danish census records where an unmarried daughter and her out-of-wedlock child are living with her parents.
Traditionally the Danish Lutheran church published banns announcing the impending marriage of a couple. At that point it was socially acceptable to live together, no need to wait for the marriage. It was not uncommon for a child to be born and the parents married each other after.
In the case of an unmarried woman bearing a child the birth record notes the birth as illegitimate. The father may or may not acknowledge the child as his. If he does, the child is given his name (the name Jensen for a child of Jens, for example). If he does not, the child is given its mother's name (if she is a Nielsen, then her child will be a Nielsen).
> "What was the Lutheran church's position on this problem? Did they shun their parishioners?" <
The church (parish) took financial responsibility for illegitimate children. This was law. The parish where the mother lived ten months prior to the birth was the financially responsible parish. Since all Danes (pretty much) belonged to the Lutheran church, this practice was akin to modern practices in which the community as a whole assumes financial responsibility for vulnerable or needy persons. Thus a Danish community as a whole or the parish in particular really can't be said to abandon an unwed mother and to the contrary made provisions for her. (remember a parish had a civil function as well as a religious function)
Sometimes we see in the records an illegitimate child who is living with grandparents or with a foster family rather than the parents, even though one or both parents are living. Somehow a provision was made for the child, if not with the parents. This life was maybe good, maybe not. Like the children of the Orphan Train in America, some foster or adopted children were beloved, others were taken in as free workers.
Of course one cannot see shock or shunning in records, which is why a really detailed knowledge of social history is needed here. I can guess it would be possible for a family to be shocked and disown a pregnant daughter, since individuals vary. My parents, children of Danes who immigrated early in the twentieth century, were raised to believe that pregnancy outside of marriage was morally wrong. (Never mind that almost any family has experienced this.) Nobody got kicked out of the family, though.
> "Would a man marry a pregnant woman to protect her reputation and accept a child that was not his?" < Well my guess is that would depend on the man. Certainly that's one of many possibilities. A child's birth record tells the story of who the father is. Genealogists can only accept what the birth record says. People do lie - but only a DNA test can sort that out.
> "I have a great grandmother, born 1868 in the USA, whose paternal parentage is questionable possibly?" < I think it would be harder for an unwed, pregnant great-great grandmother to get on a ship than it would be to stay home or near home. Immigration took courage in any case.
Most often young women who traveled alone were traveling to friends or family. A young woman could make a connection with a Danish man and travel to join him, with plans to marry. He wouldn't have to be the biological father of her child. Also, US immigration officials were not keen to admit unwed women. In 1868 a single, pregnant woman would be assumed to be unable to provide for herself and her child. Since she was apt to become dependent on American charity, she would not be allowed into the country. The solution, often, was for a friend or family member to meet the woman and claim responsibility for her.
In summary, though I can guess at disapproval of an unwed mother by her family, friends, and/or community in Denmark, I have never heard of conditions so repressive that a woman would flee her home to escape. A woman traveling to America is more likely to be traveling toward a good life (as with most immigrants) rather than fleeing an unbearable life.
So that's my two cents worth. Again, I'm no expert. And those are general comments that may not tell the story of your ancestor's situation. The Denmark General forum might have more facts and knowledge of the topic.
BTW the Danish Constitution of 1849 reflects a significant shift in Danish culture toward de-stigmatizing vulnerable people in society:http://books.google.com/books?id=TXzFqxVM5oYC&pg=PA75&am...