Good evening plattman, WOW here goes:
1) You state: No neutral and no ground just a black and white.
FYI If its the old two 2 wire system of 120 Volts THERE IS INDEED A NEUTRAL EVEN IF NO GROUND....The Neutral is actually a GrounDED Conductor. Standard (you may or may not have) Black is the HOT/LINE UnGrounded Conductor,,,,,,,,White is the Grounded Conducotr aka Neutral,,,,,,,,,,,,One wire is Hot/Line other is Neutral with 120 VAC Line to Neutral IFFFFFFFFFFFFFF ALLS CORRECT !!!!! with or without any ground
2) Would like to install a new GFI........... Fine no problem go for it even with the old TWO wire (Hot & Neutral) system !!!!!!!!!!!!!!!
FYI a GFCI receptacle can still operate and provide ground fault protection WITHOUT A GROUND WIRE. The ONLY two wires that pass through its Torroid Sensing Coil are the Hot/Line UnGrounded Conductor and the Grounded Conductor Neutral and if theres as litle as 0.005 amps of fault current IT WILL TRIP with or without any ground wire...................HOWEVER any test feature that works by passing 0.005 amps of fault current via a traditional modern Equipment Grounding Conductor WILL NOT WORK if theres no EGC present BUT GFCI PROTECTION FUNCTION CAN STILL WORK
3) and could run a bare or insulated single strand to a nearby water line.
NOOOOOOOOOOOOOOOOOOOOOOOOO Thats NOT necessary to make the GFCI work and is NOT kosher I do NOT advise such and its in violation of the NEC when and where I last practiced power distribution engineering.............
NOTE nor do I recommend any BOOTLEG GROUND made by attaching a ground wire to the Neutral
4) Would the GFI function with this type of ground on a simple circuit.
YES it will still provide ground fault protection WITH OR WITHOUT any wires to a water pipe or elsewhere and again I would NOT run any wires from it to a water pipe !!!!!!!!!!!!
NOTE Im long retired from power distribution and NOT up on the latest codes so if this is incorrect hopefully some other fine sparky gents can add to or correct this PLEASE DO SO I may well have missed something grrrrrrrrrrrr my bad
SUMMARY its my opinion it will still work (provide GF protection even if some test function cant work) WITHOUT any ground wire and I would NOT run any wires to any water pipe nor rig a Bootleg Ground See if others agree, usually you get a ton of responses to an electrical question ...........
PS In addition to kithen and bath locations, receptacles located outdoors or in shops or pole barns etc with concrete or dirt floors REQUIRE GFCI protection !!!!!!!!!!!!!!!!!!
John T Long retired EE but believe this is still true. When in doubt where fire and life safety are at risk do not take my word for this consult professional electricians and engineers and the NEC where applicable !!!!!!!!!!