The Spaniard, who had enjoyed success in Germany where he coached Bayer Leverkusen to a first Bundesliga title without losing a game, was unable to replicate that with the team he once graced as a player and left by mutual agreement on Monday.
"This stage of my career has come to an end, and it has not turned out as we would have liked. Coaching Real Madrid has been an honour and a responsibility," Alonso wrote on Instagram on Tuesday.
"I am grateful to the club, the players and, above all, the fans... I leave with respect, gratitude and pride in having done my best."
Alonso's tenure was marred by internal discord, with reports of clashes with senior players, including co-captain Federico Valverde and winger Vinicius Jr.
Real's poor performances under Alonso included humbling losses to Paris Saint-Germain in the Club World Cup, to Atletico Madrid in LaLiga, and to Liverpool and Manchester City in the Champions League.
The 44-year-old initially built a five-point lead for Real at the top of the LaLiga standings, but a string of poor results in November and December included one win out of five league matches, as champions Barcelona surged ahead by four points.
Real will visit Albacete for a Copa del Rey clash on Wednesday.