Conway will be replaced in the Test squad Mark Chapman, who will join-up with the team in Hamilton ahead of the third and final Tegel Test.
"Family comes first in this environment and we're all really excited for Dev and his wife Kim to welcome their first child," said coach Gary Stead. "Mark was with the Test squad in India recently and returned to the Plunket Shield by scoring an impressive 276 - so it's a good time for him to be joining us."
Conway managed scores of just 11 and 0 during New Zealand's 323-run loss to England in the second Test in Wellington, with New Zealand suffering their second straight defeat to England to drop out of contention of reaching next year's ICC World Test Championship Final.
The third Test between New Zealand and England commences in Hamilton on Saturday.
