The four-time Olympic champion - riding as part of his build-up to the 2016 Olympics - pulled out around 26km from the finish in cold, damp conditions.
"I'm gutted not to join you into the weekend but Rio needs to take priority," said Wiggins.
Groenewegen, 22, outpaced Australia's Caleb Ewan in a bunch finish to win.
The Dutchman, who earned a 10-second time bonus, completed the 186km stage between Beverley and Settle in five hours, nine minutes and 11 seconds.
Wiggins - the 2012 Tour de France winner, who turned 36 on Thursday - was riding for his Team Wiggins outfit.
He added: "Great atmosphere in Yorkshire today. It was a tough day's racing so thanks to everyone who braved the weather and came out to support. The boys and I really appreciate it.
"Good luck to everyone over the weekend."
JLT-Condor's Chris Lawless was the top British rider in seventh, while One Pro Cycling's Pete Williams will wear the pink King of the Mountains jersey on Saturday.
Team Katusha's Sven Erik Bystrom was left with minor facial injuries after a multi-bike crash outside Beverley that the Russian team claim was caused by a cattle grid.
Saturday's stage two takes place on a mostly flat 136.5km route between Otley and Doncaster that is likely to favour the faster riders.
Sunday's final stage, a 198km route from Middlesbrough to Scarborough, contains six categorised climbs compared to Saturday's three, and will favour the climbers in the peloton.
World champion and home favourite Lizzie Armitstead heads the field for Saturday's one-day women's race, which follows the same 136.5km route the men will tackle later the same day.
The winner of the women's race gets £15,000 - more than the winner of the men's race and £14,000 more than 2015 champion Louise Mahe.
Tour de Yorkshire 2016
Stage one result.
1. Dylan Groenewegen (Ned) Team LottoNL-Jumbo 5hrs 9mins 11secs
2. Caleb Ewan (Aus) Orica GreenEDGE Same time
3. Nikias Arndt (Ger) Team Giant-Alpecin
4. Thomas Boudat (Fra)  Direct Energie
5. Danny van Poppel (Ned)  Team Sky
6. Floris Gerts (Ned)  BMC Racing Team
7. Christopher Lawless (GB)  JLT Condor
8. Karol Domagalski (Pol)  One Pro Cycling
9. Dion Smith (NZ)  One Pro Cycling
10. Bert Van Lerberghe (Bel)  Topsport Vlaanderen - Baloise
Sir Bradley Wiggins dropped out of the Tour de Yorkshire during the first stage as Team LottoNL-Jumbo's Dylan Groenewegen took the overall lead.