Question
sap runs from limbs that were broken due to ice will it stop getting all over car

Answer
This is normal for river birch. They produce a large amount of sap during the spring and if there is a wound the sap will leak out. As the weather warms up and the sap begins to flow, the trees will appear to bleed from the pruning cuts. While not a concern as far as the overall health of the tree, it can be unsightly and the good news is will stop when the wound heals. When a birch tree is bleeding about all you can do is keep it well watered so the moisture loss is minimized. Commercially available pruning paints will not  slow the flow of sap from a bleeding wound and will slow down the healing process.  
I would suggest that the broken limb is still attached to prune it off and the wound will heal faster. Sorry, You may have to put up with the leaking until the wound heals. If the weather turns dry I would water the tree since it is losing water through the wound.