The revelation makes Rodríguez the first national beauty queen to speak publicly about being homosexual. The model's intimate picture finds her lying in bed with Klein.
"Romeo and Juliet," she captioned the image.
Rodríguez's followers responded to the photo with messages of support, encouragement and gratefulness. Days later, she posted a message thanking fans for their comments on the image sharing site.
"I published [the photo] quite naturally and impulsively," she captioned the message. "I appreciate the outpouring of support and even more to rejoice in my happiness. Thank you!!"
Rodríguez, the only woman to have won the Miss Spain pageant twice, told Spanish newspaper La Opinión de Tenerife she does not "regret what I did because I'm happy with what is going on in my life."
In a separate interview, Rodríguez, 24, told Costa Rican newspaper La Nación "there is no gender in love," adding that she met Klein through mutual friends last year and has been seeing her for six months.
Asked if she felt like she was breaking stereotypes by being the first pageant queen to come out as gay, Rodríguez said she's surprised her revelation made the news, but understands why she's the first woman in her position to come out.
"More people will [come out]" she predicted. "I don't encourage people to come out. To each their own. What do I want to encourage people who are repressed to do, is to not feel that way. Don't try to live a double life, to hide. If you don't want to come clean, don't share anything [on social media], but don't hide from yourself."
Rodríguez was crowned Miss Spain for the first time in 2008 at the age of 17, won the pageant again in 2013, and was last year's Miss Universe runner-up.