Counter Emerge Spell
Your opponent was incorrect, you can counter creatures with Emerge. All Emerge does is gave the spell an alternative way of casting the spell, but you are still casting it.
Your opponent may have gotten confused because even if you counter their emerged spell the cast trigger will still occur, and their creature is still sacrificed. In order to stop that you need a card like Summary Dismissal that specifically counters abilities.

702.118. Emerge
702.118a Emerge represents two static abilities that function while the spell with emerge is on the stack. “Emerge [cost]” means “You may cast this spell by paying [cost] and sacrificing a creature rather than paying its mana cost” and “If you chose to pay this spell’s emerge cost, its total cost is reduced by an amount of generic mana equal to the sacrificed creature’s converted mana cost.” Paying a card’s emerge cost follows the rules for paying alternative costs in rules 601.2b and 601.2f–h.
702.118b You choose which creature to sacrifice as you choose to pay a spell’s emerge cost (see rule 601.2b), and you sacrifice that creature as you pay the total cost (see rule 601.2h).