De la Fuente must cut three players before June 7th for the tournament in Germany on June 14th.
Barcelona midfielder Fermin Lopez and Real Betis' Ayoze Perez also get their first-ever international call-ups.
De la Fuente left out various players who have been involved in the last few years, including Marco Asensio, Gerard Moreno and Pablo Sarabia.
Yamal has burst into the limelight this season with Barcelona, along with centre-back Cubarsi.
The coach said he had included three extra players to account for potential injuries before then, with Real Madrid competing in the Champions League final against Borussia Dortmund on June 1.
"It's a guarantee against any set-back that could happen in the Champions League final, or in training, that would generate a problem," said the coach.
"They all have the same chance (of going), we will minimise risks, we have very versatile players."
Spain, who won the European Championship in 1964, 2008 and 2012, reached the semi-finals of the last edition.
They are in a 'group of death', facing Italy, Croatia and Albania in Group B.
Before the tournament Spain face Andorra on June 5 and Northern Ireland on June 8 in friendlies.
Spain's 29-man squad list:
Goalkeepers: Unai Simon (Athletic Bilbao), Alex Remiro (Real Sociedad), David Raya (Arsenal)
Defenders: Dani Carvajal (Real Madrid), Jesus Navas (Sevilla), Pau Cubarsi (Barcelona), Robin Le Normand (Real Sociedad), Dani Vivian (Athletic Bilbao), Nacho Fernandez (Real Madrid), Aymeric Laporte (Al Nassr), Alex Grimaldo (Bayer Leverkusen), Marc Cucurella (Chelsea)
Midfielders: Rodri Hernandez (Manchester City), Martin Zubimendi, Mikel Merino (Real Sociedad), Marcos Llorente (Atletico Madrid), Fabian Ruiz (Paris Saint-Germain), Pedri, Fermin Lopez (both Barcelona), Alex Baena (Villarreal), Aleix Garcia (Girona)
Forwards: Alvaro Morata (Atletico Madrid), Joselu (Real Madrid), Nico Williams (Athletic Bilbao), Dani Olmo (RB Leipzig), Lamine Yamal, Ferran Torres (both Barcelona), Mikel Oyarzabal (Real Sociedad), Ayoze Perez (Real Betis)