The final Test will take place in Hamilton from December 14 to 18.
Conway, who will remain in Wellington, leaves a significant void at the top of the order following New Zealand’s 323-run defeat in the second Test.
Middle-order batter Mark Chapman has been called up as Conway’s replacement.
Head coach Gary Stead expressed full support for Conway’s decision, emphasizing the importance of family.
"Family comes first in this environment and we're all really excited for Dev and his wife Kim to welcome their first child," said 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," he added.
The 33-year-old's absence might pave the way for Will Young, who impressed during last month's tour of India by scoring 244 runs at an average of 48.40.
Despite his strong form, Young has been sidelined in this series to accommodate Kane Williamson’s return.
Williamson, currently leading the batting charts with 195 runs at an average of 48.75, has demonstrated his value since recovering from a hamstring injury.
It is pertinent to mention that England secured an unassailable 2-0 lead in the three-match Test series with a dominant 323-run victory over New Zealand at the Basin Reserve on Sunday.
Joe Root's masterclass, highlighted by a creative century, combined with England's sharp bowling attack to leave New Zealand struggling, despite Tom Blundell's valiant hundred.
Root added 100 runs with skipper Ben Stokes in just 79 balls before England declared at 583, setting a massive target.
New Zealand's chase faltered early, with Chris Woakes dismissing Devon Conway and Kane Williamson cheaply. Brydon Carse and others dismantled the top order, reducing the hosts to 33/3.
Blundell's counterattacking century, supported by Nathan Smith, provided a brief fightback, but England's bowlers, led by Stokes, wrapped up the innings for 280.
Harry Brook was named Player of the Match for his explosive contributions with the bat, scoring 123 and 55 in the two innings.
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