For more than a century, Russo-Turkish relations could be described as "the Emperor's march on Constantinople." Especially in 1878, when Turkish power was so badly broken, why did the Russians refrain from seizing the city? Why didn't the vast significance of New Rome to such a deeply Orthodox nation overrule the foreign policy risks involved? And why was no further attempt made?
Because Russia, as one of the Great Powers, was very aware of the importance of maintaining the balance of power in Europe. Britain, France, Austria, Prussia, and Russia were the Great Powers in Europe in the 19th century, and they operated under with the understanding that no one country should make significant gains on the continent without the rest getting a piece of the pie (see the partitions of Poland for an example of this). So obviously the conquest of the Ottomans by one power would have been a big no-no in the eyes of the rest. However, even capturing just the city of Constantinople would have been seen as WAY too aggressive expansionism by the other Great Powers (expansion, of course, was a good thing at the time- but do it in your colonies, not mainland Europe! Think of the balance) and would have been met with immediate war to beat Russia back into line. The reason for this is that who controls Constantinople controls the Black Sea, which is incredibly important economically and militarily. And European Powers tend to get pretty bent out of shape when they're worried about your ability to control the Black Sea. For instance; The Crimean War.
TL;DR As much as Russia may have wanted to capture Constantinople for its cultural importance, they knew it was too strategically important for the other Great Powers to let them get away with it because it would upset the balance of power in europe.
EDIT: I'm not sure if a source is really necessary since the question demands more a theoretical answer than a factual one, and everything I said about the balance of power and the importance of the black sea would be found in even the most basic survey course over Russia or Europe in general in the 19th century. However, if you're interested in the the idea of the concert of europe, I'd recommend Peace, War and the European Powers, 1814-1914 by CJ Bartlett. It gives a good overview of European diplomacy and statecraft in the time period without dragging on too much.
The question is wrong. Russia never gave up. They still want it. Only Turkey is now under NATO nuclear umbrella