No. The bridge construction unit needs to cross the river in the bridge construction phase. The presence of an enemy unit would prevent this. Any enemy units need to have been destroyed/forced to retreat etc. to leave the opposite side of the river vacant.
Did this rule change in 4th edition? I ask because I only own 1st edition, and in that edition, the rule (13.3, page 9) reads that if the hex on the other side of the repaired bridge is free of enemy units, the bridge repairing unit MAY move across during the bridge repair phase of the turn, along with any other friendly units present in the bridge repairing hex - the rule makes it clear that this move across the bridge is optional, not mandatory. The units must stop on the other side. This is one of the few times when a friendly unit is permitted to move directly from one ZOC hex to another ZOC hex of the same enemy unit. Again, I don't have the newer edition, so maybe they changed that rule.