Can a LCD projector imprint an image onto a light-sensitive paper?

Yes that would work, assuming the projector can focus close enough. Getting your exposure right would be interesting.

However the final quality will be limited by the resolution of the projector.

Noritsu, a Japanese manufacturer of photo equipment designed and marketed mini-lab printers based on the LCD. Also, Electronic System Engineering (ESECO) marketed an enlarger based on an LCD display. Both project an image on standard color negative paper which develops up as a positive color print. The ESECO enlarger micro-moved the LCD display as the projected image played on the paper. This scheme greatly multiplied the resolution of the finished print. This resolution multiplication is possible because most camera digital files contain far more data than the LCD can display. Software was devised that micro-moved the LCD and at the same time the LDC displayed pixels that otherwise would be discarded. These were systems I was involved with. Kodak, Fuji, and many others produced color printing devices based on the LCD. A successful spin-off, Noritsu used a modified Texas Instrument micro-mirror array and made an impressive color printer and they marketed this device.

Almost all commercial printing on photosensitive paper in 2020 is done using either LED or laser emitting "enlargers", though most are scan line in nature that use the same array to move from one side of the paper to the other, sort of the reverse of a scanning back camera, rather than a three dimensional display that emits the entire image at once.