Will a GFCI receptacle meet the code requirement to repair an open ground circuit?

Ground is not used by a GFCI protected circuit. GFCI devices monitor the amperage on the Hot and Neutral wires. If there is a 0.006 amp difference between these two wire, the GFCI device opens the circuit. It a device used to prevent electrocution.

However, if you are replacing a two wire (Hot & Neutral) ungrounded receptacle, it is permissible to use a GFCI receptacle, but the receptacle must be labeled as "Ungrounded"

Big difference between an open ground "circuit" and an open ground at an outlet. What exactly are you trying to accomplish. Putting in a GFCI does not cure an open ground.

Will a GFCI receptacle meet the code requirement to repair an open ground circuit?
Repair an open ground on a grounded circuit, definitely no.
Be installed on an old non-grounded circuit, yes.