Thank you to all who have given experience reports, lessons learned, opinions, and advice.
I did use my new D4 for two days at the Maryland Renaissance Festival, 20 & 21 October, with no protective cover on the monitor. I did need to clean the monitor afterwards with a drop of lens cleaning solution and a micro fiber cloth. No damage is discernible.
WRT to the Plastic Monitor covers supplied by Nikon for the D200 and the D300: The D200 Manual says: "A clear plastic cover (the BM-6 LCD monitor cover) is provided with the camera to keep the monitor clean and to protector it when the camera is not in use. ...." The D300 manual says the same about the BM-8 monitor cover. So, Nikon seems not to intend these covers to be installed on the cameras during use. I didn't know that. I didn't learn it in 6 years of use.
I will be careful to keep diamond tipped cutters and cut diamond rings away from the monitor. I'll warn any ladies with rings who might who might need to take photos with it.
I intend to contact Giottos USA to ask if (when) it intends to produce a Professional Monitor protector specifically for the D4. It makes monitor covers for specific cameras, and for many models it provides an "ALSO FITS" list. D4 is lumped in with the Nikon L110, D800, and D3200 in the "ALSO FITS" list for the "Giottos SP8306 AEGIS Professional Glass Optic Screen Protector- 12 Layer" (which is made specifically for the Canon EOS Rebel XSi).
Giottos makes specific protectors for the following Nikon Bodies: D50; D70S; D200; D80; D40/D40X; D2X/D2Xs; D300 (ALSO FITS D300s, D700, D90, D3000, D7000); D3 (ALSO FITS D3x, D3s); and V1.
A review of this particular protector, the SP8306: "The is a very high quality product and very well made. I purchased this screen for use on the Nikon D4. Although this LCD protector was made for some Canon models, there was a sticker added that it would fit the D4. It fit perfectly in the width but was slightly shorter in the height. The adhesive border on the outside tends to mask some of the screen area such as the scroll bar on the right in the menus. I'm still using it as it does add some protection but would have been happier if it was a better fit. One has to be very careful installing it."
The reviewer's comment about the scroll bar on the right of the menus being masked makes me think that a bit more width is needed too, but maybe the required adhesive width makes this masking inevitable even if the glass area is a precise match to the Nikon Monitor glass area.
Perhaps if Giottos USA hears from enough D4 users it will produce a D4 specific model. Would a D4 protector be perfect for the D800 and the D600 too? I don't see the D600 mentioned by Giottos.