No
Wrapping clothes around it will not safely protect it.
Find yourself some large-bubble bubblewrap and wrap it with the bubbles side down toward your monitor, flat membrane side out toward the exterior. That causes the flat membrane to stand off away from the monitor an inch or so.
If you wrongly put the flat membrane side against the monitor, a damaging object can easily come in between the bubbles and contact the monitor.
I'd wind 2 layers on there & tape it off. That'll make it bulky. Good. That's insurance against damage.
Carry on & put it in overhead luggage. Watch that no late boarders dump other stuff on top of it in the oh luggage bin.
If you can't carry it on, then find a heavy cardboard carton to put your bukly bubble wrapped monitor in, fill the empty space with more bubblewrap, tape it off, check it as luggage. What else can you do?