Do I have to replace one extracted tooth?

The biggest concern that I would have is that in areas of missing teeth you can have adjacent teeth move into the space. Usually the top teeth will grow out of the bone and into the space, and the bottom teeth usually start tipping towards the front of the mouth, creating future complications and potentially unnecessary expenses. Also consider that in areas of missing teeth the bone tends to shrink over time. This can also make future implant treatment more complex and perhaps more costly. 

I'm sorry to hear that you had to remove your cracked tooth. I will make a confession, I too had to remove a cracked tooth about a year ago, and I too knew I had a decision to make. As a cosmetic dentist, I knew that the right thing to do was to replace that tooth, in some manner, as soon as possible. When you lose a tooth, for any reason, a variety of things start to happen. First, the bone in that area will start to go away. Have you ever seen an older patient with their dentures out? It looks as if their face is collapsed. Ideally, when the tooth is extracted the dentist will also place some sort of grafting material (socket preservation) to help slow that down or avoid it. Second, the teeth next to and above the extraction site will start to move. Even though we don't feel it, teeth are actually always shifting, and they depend on the other teeth around them to keep them in place. Third, slowly you will start to have changes in your facial soft tissue (your face will slowly start to go in at that spot).
When faced with the replacement of a lost tooth, a dental implant is the gold standard and should be everyone's choice. A dental implant replaces the missing tooth without touching the neighboring teeth. A dental implant also preserves the bone in the area of tooth loss. A dental implant will NEVER get a cavity. Unfortunately, the cost of an implant is usually a deterrent for patients. There are many patient financing options that I offer in my practice, as do most dentists, that help spread that cost out over a few years even. You can also visit a dental school (I believe you are near Stony Brook) or a hospital dental clinic, where an implant would be done at a much cheaper price by doctor getting their implant training. Your next choice would be a fixed bridge. It would be something that you would not have to take in or out, but, yes, you would have to shave the teeth next to the space. Also, you would have to use special threaders to keep it clean, and it can also get cavities in the future (if you don't maintain it well). Your last, and most INEXPENSIVE option would be to do a single tooth flipper. It would be a small removable tooth that you can place in that area. Though it would serve to "fill in" the missing tooth in terms of looks, the true purpose of it would be to maintain the space and keep the other teeth from shifting. It will also help with chewing, as you would not have an empty space there.
No matter what you choose, you should replace that missing tooth. I hope you can see that you have some more options, and why you should do something.
Best of luck, Dr. Alvarez

Hi:Normally when a tooth is extracted with an exception of a wisdom tooth, if it is not replaced it can cause shifting of the opposing tooth and adjacent teeth and ultimately effect your bite.  It is always best to replace the tooth with either a removable partial, bridge or implant.Best,Dr. Maddahi