Ronald Koeman's side earned a 3-1 win after extra time in the semi-final clash at Estadio D Afonso Henriques in Guimaraes, to earn a place in the showpiece event against Portugal on Sunday.
The Three Lions took the lead via a Marcus Rashford penalty in the first half, but the Oranje took control of the match thereafter and were rewarded with an equaliser in the 73rd minute.
Matthijs de Ligt's header ensured that the contest would last an extra 30 minutes and individual errors ended up costing England as fatigue started to creep in.
John Stones was guilty of dwelling on the ball in his own box in the first period of extra time, and Memphis Depay capitalised before the ball was eventually turned into his own net by Kyle Walker.
Quincy Promes then rounded off the scoring in the 114th minute, after Depay intercepted a loose Ross Barkley back pass and laid the ball on a plate for his team-mate to volley home.
Rice, who saw 105 minutes of action, confessed England were the architects of their own downfall post-match, as he told ITV Sport: "It’s gutting, it really is, it really is gutting.
“It’s tough. I’m lost for words. First half, we went out there and tried to play our stuff and we obviously had the lead, which I thought was deserved.
“Holland didn’t really create any chances at all in the first half and then, the second half, we just kind of sat back and we didn’t do what we did in the first half. We didn’t get up in their faces and make it difficult.
“They obviously got the equaliser from a corner, which is gutting because we work on them day in, day out, and obviously [we had] that VAR controversy again, which was really tight and might have changed the game a bit.
“Holland won and we cost ourselves the game with two mistakes and I’m gutted about it.”
The West Ham midfielder went onto defend Stones for his costly mistake, urging his colleague to pick himself up quickly and move on.
“Look, he is a great player, John Stones, and that’s his game, he takes risks," Rice added. "It doesn’t matter that he’s done that, we’re all behind him.
“We all know we make mistakes in football. The manager backs him, we all back him and it’s one of those things in football where, if you make a mistake, you’ve got to pick your head up and go again.
“I’ve made plenty in my time and it’s one of those where you’ve got to lift your head and go again but it was tough to take.”
England have the opportunity to get back to winning ways with a third-place play-off against Switzerland on Sunday, who were beaten 3-1 by Portugal in the first semi-final on Wednesday.