Generally speaking, the answer is no, sharing the same language doesn't play a sound role in the likelihood of war.
Richardson's (1960) Statistics of Deadly Quarrels provides the
earliest systematic treatment of the impact of cultural similarity
on international conflict. Analyzing more than 300 wars and disputes
from 1820 to 1929, he finds that, in the main, common language did
not predict either increased or decreased bellicosity among warring
parties (pp. 223-30). However, for the major language groups that he
studies (N = 13), Chinese dyads tended to be less war prone, whereas
Spanish dyads were more likely to fight one another. Contrary to the
belief of the advocates of universal languages, such as Esperanto,
that common language across states would increase the likelihood of
peace through more effective communication, similarity or difference
in language between states seemed to have little effect on the
occurrence of war.
(cited from here)
Most of the contemporary conflict studies prefer investigating the effect of ethnicity on the likelihood of armed conflicts, and though ethnicity and language are strongly related to each other, ethnicity is believed to be a better explanatory factor than any merely linguistic factors.
Nonetheless, the importance of language as a means of communication and as an identity factor makes it a convenient instrument of propaganda both for conflict escalation and for conflict resolution. An interesting investigation of Yugoslavia's case could be found here. And here is a broad discussion of the role played by language and discourse in conflict resolution.