Yes, you are correct, on the movment phase the Investigator moves back to Arkham, monsters will ignore the Investigator. If, during the immediately next Arkham Phase, you can close (and seal if possible) the gate, all is fine.
If not, monsters will notice you, & will want to discuss Napoleonic history with you (using their tentacles).
Actually, you have to deal with the monsters the next turn, period. (That is, regardless of whether or not you managed to close the gate - unless the gate closing removed the monster, of course.)