Gravity Field

No. The 1 cost does not apply because you perform the move action, not the opponent who controls the unit.

The player who performs a game action is responsible for it.^412.1 For example, Charm and Siren's Song each instruct you to move an enemy unit, so you are responsible even though your opponent controls that unit. Gravity Field's applied cost applies only when an opponent performs the move action to its battlefield.^204.4204.4.a

They must pay 1 to perform the move, but they may decline. If they decline or cannot pay, they move no unit.

For example, Angler Wyrm and Call to Battle each instruct a chosen opponent to perform the move, so that opponent is responsible for it.^412.1 Gravity Field therefore applies its cost to that move action.^204.4204.4.a Applied costs are paid as their associated game actions are performed.^204.4.b

The word "must" does not remove the option to decline an applied cost. If the opponent declines or cannot pay, they cannot perform the move action.^204.4.c The move instruction is negated rather than ignored because it cannot be followed for a reason other than an illegal target or unavailable information.^{359.3.e.6407.5}