Stem spot is not a recognized birch disease but rather a description.  The following websites give detailed birch disease descriptions.  Compare what you are seeing with these.  I suspect that you are dealing with canker rot.
1. https://apps.extension.umn.edu/garden/diagnose/plant/deciduous/birch/index.html    (Click on the trunk areas)
2. https://www.gardenia.net/disease/canker

Is it OK to keep over winter without treating?

Can the fungus, Canker  spread to nearby to Oak and Maple trees?

I don't want to use chemicals secondary to risk to our well.

Plan is to keep over winter if the fungus doesn't bother other trees and birds.

Can you advise me?

 Thanks!

Mary D Hall

Since canker is caused by a number of different fungi, it is impossible to predict if it will infect your maple and oak.  Those trees are susceptible to some cankers. As the second website above shows there are lots of cankers and it lists which trees are susceptible to which fungi. These organisms are wind blown so your other trees will likely get some of these fungi but they may or may be susceptible.  See:
https://mortonarb.org/plant-and-protect/tree-plant-care/plant-care-resources/canker-diseases/

If these are high value trees you can send a specimen to the Plant Disease Clinic for analysis as to which fungus is involved and whether the other trees are susceptible. See:  
https://pdc.umn.edu/

The following article is quite technical but it gives some good diagrams on how pesticides can enter wells. 
https://www.usgs.gov/special-topics/water-science-school/science/pesticides-groundwater