Hey Jared,
Both 220/240 V and 110/120 V are single-phase electricity. The fact that in the U.S. you usually find split-phase electricity with two hots does not change anything on the fact that such a supply, when connected between the two hot legs, are single-phase electricity. The VFD that you connect to the two hot legs is one for single-phase electricity.
The fact that a device on 240 V single-phase draws half the current than on a 120 V voltage source, the same power (in Watts) assumed, has its cause simply in the formula P = U * I.
And it is true that half the current requires half the wire cross-section area diameter and half the fuse rating, but this does not apply to American Wire Gauge (AWG) because half a AWG wire gauge does not mean half the wire cross-section area. Using half the wire strength on half the current is true for wire gauges given in wire cross-section area, i.e. IEC 60228. The reason that a double cross-section area fits double the current comes from the fact that it has exactly half the resistance. This linear relationship does not apply to AWG.
See also: Comparison chart of AWG (blue), SWG (red) and IEC 60228 (black) wire sizes (hover mouse over a field to see dimension)
For computing the power draw of a spindle/VFD see here.
As a VFD is a AC-to-DC-to-three-phase AC inverter, so the AC on its input is converted to DC anyway, and it does not matter if you feed it with single phase or three-phase power. It will not affect “cleanness” of the electricity. What will strongly affect the cleanness of the output current are AC and DC reactors and RF filter chokes that are available for VFDs. Also I would recommend to always use a input line filter matching the input current and voltage of the VFD (usually available as accessory, see VFD manual, Section 5 “Inverter System Accessories”)
The difference between single-phase VFD input und three-phase VFD input is that the formula for three-phase power is P=U·I·√3 so you have the current coming over three wires, and each provides an AC current whose phase is shifted by 120° towards the next. This means if you have three-phase supply in your home (like it is usual in Europe), you can use a much weaker wire gauge/wire cross-section area to supply your VFD than it would require on a single-phase (or U.S. 240 V split-phase current which is also single-phase), but you need two more wires then.