Why is the resistance of an ideal voltmeter infinite?
Your reasoning is correct if you take “voltmeter” to mean galvanometer (a device which deflects a needle using a coil in a magnetic field). But “voltmeter” just means a device which measures voltage, and we can imagine taking that measurement without disturbing the circuit.

The whole idea of an ideal instrument of any sort is to ignore the limitations of a physical implementation of the concept; there does not have to be a possible way to do it.



That said, there are ways to measure DC voltage with a nearly infinite input impedance.


The electrostatic voltmeter or mechanical electrometer. An electrostatic voltmeter has a needle and pivot like a galvanometer, but instead of a coil, has a pair of shaped plates, much like a variable capacitor, which are electrostatically attracted by the voltage across them.

The electrostatic voltmeter electrically resembles a capacitor, so its input impedance is infinite at DC (if you ignore leakage across insulators). If you remove it from a circuit, the needle will keep its reading unless you short the meter terminals!

In this type of meter the force is very small compared to a magnetic type and so it only works well for voltages in the tens to thousands of volts, and takes a long time to settle. Compared to a galvanometer, it is less practical to be adapted to measure current or have multiple voltage ranges.
A null voltmeter or (obsoletely) potentiometer uses the bridge principle to measure voltage. In manual operation, an adjustable reference voltage is connected through a galvanometer to the circuit being measured, then the reference voltage is adjusted until the galvanometer is not deflected in either direction ("nulled"). The same principle can be implemented electronically without manual controls.

This type of voltmeter will either draw or supply (!) power from/to the circuit under test, until the nulling is done; the effective input impedance depends on how precise the null is.

(One of the original applications of these devices was measuring the voltage generated by thermocouples.)