Why can't we call dog breeds different species?
I don't enjoy the idea of comparing people to dogs, but this is a bit akin (please note the bit) to asking if people who are genetically short (or "little people", both proportionate and disproportionate dwarfism), pygmies, short people, normal sized people, tall people, very tall people, and people with acromegaly (gigantism) can be called different species. I compare because it's fairly clear to most individuals that we are all the same species: homo sapiens. About breeding, it is very likely that a reproductive specialist would be quite concerned (I certainly would be) about a woman with drarfism carrying the child of a very tall person (who had very tall parents), but not the opposite pairing.

Below is a photograph of the wedding of Tom Thumb and Lavinia Warren. That union produced no children. Her bridesmaid Minnie, (her sister), who was only 27 inches tall, also married a little person (Edward Newell). She became pregnant with a normal-sized child, but died during the birthing of a 6 pound baby (her baby died as well).



Are they a different species? No. A paleontologist (a good one) would be able to tell that the bones of a little person was actually of the same species based on certain conformations, just as I believe paleontologists today could distinguish prehistoric small cats from very large cats. Size matters, but not nearly as much as the type and distribution of teeth, the shape of the jaw, etc.  In other words, I strongly disagree with the following:


  If somehow the recognized [dog] breeds existed only as fossils, paleontol­ogists would consider them not one species but many—certainly more than the thirty-six species of wild dogs that live in nature today.


That statement is based on a fairly profound error in thinking: a cognitive bias that is mistrustful of science in general due to a lack of familiarity with the rigors involved in the scientific methods of paleontologists in particular. I say this without rancor or animus of any kind; where I settled down (in the deeply religious region), the vast majority of my children's friends' parents are creationists, and the vast majority of our friends are fundamentalists. We argue about science all the time; I know the basic arguments. I'm familiar with their websites. I personally find myself in the camp of Christian thinkers like geneticist Francis Collins, M.D., Ph.D., the director of the Human Genome Project, who understand genetics and find no threat to their faith in evolution or science. (E.g. Creationists reject all science which indicates the age of the Earth is >~6000 years.)

The truth is that the definition of a species is not that easy, even for prokariotes. Difficult as well is a pure definition of life.

Finally, about size. The domestic dog exhibits greater diversity in body size than any other terrestrial vertebrate. This is due to a small number of (or perhaps even a single) alleles: the IGF1 Allele. It's hard to argue difference of species on the basis of a small number of alleles: we allow much more variability for other alleles without wanting to classify as different species: eye color, hair color, hair length, straight vs. curly coat, degree of tail curl, ear morphology, etc.

A Single IGF1 Allele Is a Major Determinant of Small Size in Dogs
Lessons learned from the dog genome
A species definition for the modern synthesis 
Single-Nucleotide-Polymorphism-Based Association Mapping of Dog Stereotypes
Canine Genetics Facilitates Understanding of Human Biology 
How many species of prokaryotes are there? (PNAS)
Ernst Mayr and the modern concept of species (PNAS)
Hybridization, ecological races and the nature of species: empirical evidence for the ease of speciation
Is it Useful to Have a Clear-cut Definition of Life? On the Use of Fuzzy Logic in Prebiotic Chemistry