Why can't a Wheatstone bridge measure resistance of very high value?
You are mixing things up, the text talks about a galvanometer which is an electomechanical device which draws its power from the circuit in which it is used.



When the currents become too small (less than a few micro Ampere) a galvanometer cannot be used anymore as you will not see the needle move anymore.

A Wheatstone Bridge:



Is a circuit which can be used to determine the ratio of values of components. As such it is suitable also for high value resistors. What is required though is a sensitive volt/current meter in the middle.

A galvanometer is not that sensitive so a Wheatstone Bridge using a galvanometer is limited to resistor values up to around maybe 1 Mega ohm.

Replace the galvanometer with a more sensitive (electronic) meter (which could still have a galvanometer for readout) and you can use the Wheatstone Bridge also with high resistor values.

So the limitation is not due to the Wheatstone bridge but due to the galvanometer.