"Gatsu" is for months of the year, whereas "tsuki (or kagetsu)" is month-long periods. "Nichi" indeed can refer to days of the month, but "ka" is used for 10日. See the following explanation on wiki.
Japanese counter word - Wikipedia, the free encyclopedia
In conclusion, 4月10日 is read "shigatsu tōka", not "4 tsuki 10 nichi". (FYI, "yotsuki" means "four months". "Tōka" can also refer to "ten days", but neither "jūnichi" nor "tōnichi" is used.)