I have a critical problem when i connect my laptop to a projector. Colors that I have used in my website are getting washed out when it's displayed on the projector screen.
This is normal behavior for projector vs. screen. They are not the same, just as screen vs. print is not the same. You may have to do a mockup of your website in Photoshop with different colors and experiment with the projector so that the end result looks like what your website is supposed to be.
Does it look okay if you connect it to another projector / LCD screen? If so, it may be that there's an issue with the projector (if it's an older one there may be burn-in or it may be that the LCD unit is dying and showing wash-out). Check the cable - if you have a broken pair you may not be getting all of the R, G, or B information. If you can verify it's not the cable by just swapping the cable out.
If it looks washed out on another screen / projector, it may be the output port settings on your laptop. Without knowing the type and age of your laptop I can't make any definitive statements but you could try swapping your primary and secondary displays to see if it's an issue with the way stuff is being displayed out of that port.
This PDF will help you. Change the color setting in your displays. You'll understand what I mean when you read this.