2 bottom molars (18 and 19) were removed. I don't want an implant. Any chance that gap can be filled up without doing anything?

The only option other than a dental implant would be a partial; however, partials are only intended for looks not function or bone loss prevention.

If tooth #18 and 19 are completely gone and you are not interested in implants the other two options are a 4-unit permanent bridge or a removable unilateral bridge, since one of the pillar molars would need to be a 3rd molar (wisdom tooth) this option might not be a very good idea probably the partial is the best alternative for implants in your case.

The teeth are in constant movement during our whole life and is normal that when you lose a tooth at an adulthood the neighbor teeth start moving towards the toothless area and even for the upper to shift down these movements contrary of what you think will not benefit or cover the toothless area partially because they will move tilted instead of in a straight position. 

Hello, if you don’t want do implants you can do removable prostheses to replacing the teeth miss. But I recommended you do dental implant to get better results and get it when you have enough bone to do. 

Honest7213:Dr. Timmerman is correct.  Your upper teeth want contact with lower teeth.  If those lower teeth are missing, it's possible that your upper teeth will grow downward and create a new problem for you.
A removable partial denture is a good idea if you're not at all interested in replacing those missing teeth with implants.  If, however, you think you might want implants (but not now), you may be able to get by with a Temporary Transitional Partial denture (TTP), also sometimes called a "flipper."  These TTPs are not durable and not meant for long-term use, but if you're only waiting a few months or a year, sometimes a TTP is a more-affordable way to keep those top teeth from moving while waiting to have your implant treatment started.
Good luck!Kevin Fitzpatrick, DDS, MS

There are other options, fixed prosthesis, but to be able to replace 2 teeth it is necessary to make a fixed bridge with 2 or 3 healthy teeth, the healthiest option is dental implants, the following options that follow are removable prostheses.

When teeth are removed, the socket will be left behind.  A blood clot will form and the gum tissues will cover the hole, and the clot will eventually turn to bone.  However, no teeth will move to replace the missing teeth, and the upper teeth may slowly descend and may even eventually touch the gum tissues where the teeth had previously been.  If not getting implants, I suggest a partial denture.