Doubtful. The diseases that came to the Americas came on the backs of Exploration ships so a full city, especially one the size of Constantinople, is going to have all the diseases that matter(Small Pox, Tuberculosis, Cholera, a smidgen of Bubonic Plague, etc.) within it.
That logic doesn't work. The question isn't "how many people came over", it"s "how many people formed the population group Constantinople was part of, before it was transported vs how many people formed the population group those ships were part of".
Some other important factors to consider:
1)The explorers who went to America by ship, were explorers, sailors and soldiers people prone to collecting parasites and diseases for various reasons making it more likely they'd have picked up any given disease.
2)The ships carried only men, who after months at sea were anxious to seek close contact with the natives, (particularly the girls) which provided many opportunities for even diseases with low infection rates to jump across.
3)There are in many cases lesser forms of a disease, or other diseases that provide at least some resistance to the more deadly form, and these were not brought over by the historical ships, but were brought over by Constantinople (Cow pox, for example).
4)Just because a disease was present in 16th century Spain, doesn't mean it was in 15th century Constantinople.
5)The population and culture of the Amerindians in California is going to be very different than those the historical explorers encountered, and that may (or may not) produce better results when dealing with a plague.
6)Constantinople has enough resources they actually can help deal with the plagues if they want to (as well as they know how at least), something the ships couldn't do historically.
In short, while there might by massive die-offs, there also might not.
