The 31-year-old put the agony of missing out on the medal places four years ago in Beijing behind him with a medal this time around.
“It’s been a pretty difficult four years but I always believed I could come here and do well. I’m just massively relieved more than anything," Dempsey said.
“It’s been really tough. The Dutch guy had such a good start [to the regatta] and it was about coming to terms with that very quickly I was only going to be racing for second.
"I was desperate, desperate [to win silver]. Third was going to be a disaster, and fourth I don’t know what I’d have done.
“I’m just glad I could do it for all the people who have helped me. Without doubt to have all my family and friends here to do in front of the crowd couldn’t be better. This is a moment for reflection but I’m pretty sure this is not the end of the road.”
Poland's veteran Olympic board campaigner Miarczynski won bronze in his battle with Germany's Toni Wilhelm.
This was the last Olympic men's race on the RS:X board following an International Sailing Federation decision in May 2012.
Kiteboarding will replace windsurfing at the 2016 Rio de Janeiro Games.
Dutch officials have said that Van Rijsselberghe will leave for California straight away to take up kiteboarding.
The 2011 world champion board scored six firsts from the 10 qualifying races, leaving all in his wake by the halfway stage, including 2008 silver and bronze medallists Julien Bontemps of France and Shahar Zubari of Israel.
Meanwhile, the women's equivalent was won by Spaniard Marina Alabau Neira later in the day.
Tuuli Petaja of Finland won the silver medal and Zofia Noceti-Klepacka made it a double RS:X bronze for Poland.