No, the width of the freehub (thing that the cassette slides onto) is different for 7 speed vs. 8/9. If you have an 8-speed then a 9-speed will work because the overall width of the cassette is the same with 8 and 9, just the spacing between is smaller.

Also, as you move up to 9 speed, the chain gets much thinner. As a result, your chainrings (front 3 rings) are too thick and the chain will not drop down between the teeth.

Also the width of the jockey pulleys (two little cogs on the rear derailleur) will cause the same problem and the cage holding them will be too wide and cause horrible shifting.

You'll need new shifters because a shifter is designed to pull the cable a proportionate amount to the width of the cassette so one shift at the shifter equals moving the rear derailleur enough to change one gear. Since the spacing is different your shifting would always be off.