Neutral gray surfaces (paint) that do not contain advanced additives that increase gain at the expense of viewing cone do indeed project white.
First off a neutral paint is carefully crafted of pigments the major components being white (titanium oxide) and black (lamp black/ carbon) in mixing these pigments together in a base solution that tries to remain clear and not impart color shifting and also allows the paint to dry and stick to a surface some color shifting is going to happen. There was a tremendous amount of effort put forth here a few years ago using state of the art color measuring equipment and carefully analyzing the results of different combinations of white, black and binders of a given brand of paints and then adjusting the mixture with corrective pigments in the red range to achieve the closest possible neutral surface to reflect projected light off of. There could always be final correction done with a calibration of the projector to reach that perfect spot. The correction could be needed do to the individual projector ability to produce white light in the 6500k range. 6500k illuminate was what all the experimentation was done around and is generally regarded as the correct illuminate for projection much like sunlight.
To simplify the idea of what goes on in a paint you have to know the pigments do not combine. They are microscopic particles evenly mixed together. In the case of white they reflect almost all the colors in the light striking them and absorb very little. Black pigment is just the opposite absorbing most of the spectrum reflecting little. If we add a red pigment it will absorb all but the red light. But keep in mind each is doing its own thing.
Now picture a football field covered in pool balls 90% white 9% black and 1% red all mixed randomly and from close up you see each ball as the color it is. Now go to the upper most seat in the stadium and looking at the field your vision can't discern a single ball but sees them as a solid neutral gray surface. You shine a light at that field the white is still projecting white back to you, what you see as gray is white with the intensity turned down is all. It's exactly the same thing you would see with a projector with a neutral density filter placed over the lens, and that's very commonly done with overly bright projectors and or small screens.
To answer the question then yes you will need to do a calibration if your old screen was neutral white and you make a neutral white screen of a lower intensity neutral gray and turn the brightness of the projector back up you will have the same result you had before except the gray screen will act on ambient light attenuating some of it. Lowering your black reference and enhancing your perception of black. Perceived contrast is another story. As is adding gain. Like how a neutral gray can reflect true white, adding gain is the same thing as adding brightness in many ways. Directional gain does have the benefit of reflecting less into the room close to the screen and causing more of it to go deeper into the room. It also has the ability to act on stray light coming from the side and doing one of two things depending on the type of gain medium it is, skip the light off the screen across the room or reflect it back at the source. Both keep it out of the viewers eyes and preserve contrast.
IMO more can be done to improve the ambient light viewing experience thru placement of the light sources and direction along with wall and ceiling and floor treatments closest to the screen than imparting gain into the paint. Many disagree with this and many people don't have total control of the room to do such things.
Below is a screen shot taken from a screen that is totally black to the eye. It was also very small about 30 inches in size and with a very bright projector. The reason I did this was to show how white can be projected off even the darkest of neutral grays given enough lumens. I might also mention the room was in a very bright state of ambient light at the time. So this was an extreme test to show the abilities of a simple neutral gray.