If I use Gravity Field on a Battlefield and I then play Blitzcrank, Impassive on this Battlefield, does the opponent have to pay the cost of the move, or can he deny the move because he doesn't want to pay the cost?

No — the opponent pays nothing and can't refuse the move; Blitzcrank just pulls their unit over.

Gravity Field's "opponents must pay [1][A] to move one or more units to this battlefield" is an applied cost, and applied costs attach to the player performing the move (Rule 204.4). Blitzcrank, Impassive's "when you play me to a battlefield, you may move an enemy unit to here" is your ability and you're the one doing the moving — your opponent isn't taking a move action at all, so there's no cost for them to pay and nothing for them to decline (Rule 204.4.c only lets the player who would perform the action bow out).

Gravity Field only taxes your opponent when they move units there — their Standard Move, their Ganking move, or their own effect-based move — and in that case yes: if they can't or won't pay the [1][A], the move can't happen at all.

They do still get a priority window while Blitzcrank's play trigger sits on the chain, but the only way to stop the move is to counter the trigger or remove Blitzcrank before it resolves — not by refusing to pay.