If VGS = +3.7V your pMOSFET will be OFF. If it's -3.7V, then your pMOSFET will be ON (see last row of your table).
To be clear: VGS = +3.7V means that, with your data (the source is at 5V), the gate is at 8.7V with respect to ground. With VGS = -3.7V, it means that the gate is 1.3V with respect to ground.
If you apply 3.3 V at the gate (I assume with respect to ground), then VGS = -1.7V which is larger (in absolute value) than the absolute value of the threshold voltage, i.e. the MOSFET is ON. However, it might dissipate more because of the large Rds. There is no zoom of the output characteristics (see vishay datasheet) for such low current values (4A), but it seems to me that with VGS = -1.5, you'll have a VDS of about 100mV or so (you'll dissipate 400mW).