If a husband tells his wife our relationship is over and he sleeps in a different room, is this considered to be an Islamic divorce?

Saying our marriage is over could indicate an intention of divorce but could also be a verbal warning only.  Sleeping in a separate room for months without declaring a clear intention of divorce, does not start a Talaq process. The Talaq process starts only from the time the husband says to his wife you are divorced and her i’dah (waiting period) starts then as well. If the couple don’t get back together within 3 months (i’dah-waiting period), the divorce is final and they need to separate from each other as they are no longer a husband and wife.