Does using a macbook with an external monitor damage the health of the battery? (MacBook 16inch 2019)

Does using a macbook with an external monitor damage the health of the battery?

No.

Use an external display with your MacBook Pro - Apple Support

Using an external monitor or leaving the computer plugged in will not damage the computer or the battery. Leaving the computer plugged in reduces the rate of accumulation of battery cycles.

Your computer performs best when connected to AC power. It can use the full output of the Power Adapter AND when doing especially challenging work will also freely "borrow" power from the battery. In some cases, the charged state may even decline during stressful work.

When used only on battery, your computer has no extra cushion of power, and will perform more slowly. However, for ordinary non-stressful tasks this may not be objectionable (possibly not even noticeable.)

In general, you should ALWAYS connect AC power when it is possible to do so, and only run on batteries (which will be somewhat slower) when no AC sources are at hand. There are three micro-controllers cooperating on battery and charging issues, and your Mac will NEVER over-charge.

An external power supply that provides "USB Power Delivery" (like these displays) can not 'force itself' on your Mac. The Voltage and Current are delivered only after your Mac requests and the charger agrees to supply power under certain controlled conditions. The computer is in control of the entire process.