This is like trying to beat rocks with scissors in rock-paper-scissors. Greater Fire Elementals have 90 Melee resistance, but only 25 Elemental resistance. Trying to beat them down in melee can work if you have a hugely overpowering force, but that's really not a good strategy.
Fire Elementals have poor Elemental resistance and very high Melee and Missile Resistance. Earth Elementals have very high Elemental resistance, moderate Melee resistance and pretty good Missile resistance. Because Earth Elementals are restricted to Melee attacks, you can kite them or offer up durable units for them to beat on while you grind them down. Fire Elementals, on the other hand, do Elemental damage instead of Melee damage, and they are ranged. Strategies for taking on Earth Elementals are Fire Elementals are pretty much the exact opposite.