The June 2 clash between Wales and South Africa in Washington DC falls outside World Rugby's June Test window, and players located in England will not be made available by Premiership Rugby for the Springboks fixture.
Premiership Rugby Limited, which represents the English top-flight clubs, says the players are therefore not eligible to play in that match.
Wainwright gets his opportunity due to the dislocated shoulder suffered by Navidi in Cardiff Blues' Challenge Cup final win over Gloucester.
Wales' contest with the Springboks in Washington DC sits outside World Rugby's designated worldwide window - which begins on the weekend of June 9 and runs until June 24.
The three Saturdays within the June worldwide window are 9, 16 and 23 June, with Wales having no extra game in this period.
Tomas Francis (Exeter) and Luke Charteris (Bath) have been pulled from Wales's summer tour squad after their Premiership clubs refused to release them early.
"The situation is very clear for players playing in England going forward and heading into a RWC [Rugby World Cup] year".
All three players will take no part in Wales' games against South Africa and Argentina and have been immediately replaced in Warren Gatland's touring party.
Forwards: Rob Evans, Alun Wyn Jones, Nicky Smith, Elliot Dee, Ryan Elias, Rhodri Jones, Samson Lee, Dillon Lewis, Adam Beard, Bradley Davies, Seb Davies, Cory Hill (co-captain) James Davies, Ellis Jenkins (co-captain), Ross Moriarty, Aaron Wainwright, Aaron Shingler.
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