Franklin, otherwise known as the Queen of Soul, died aged 76 on Thursday at her home in Detroit, Michigan, following a battle with cancer.
Mr Obama and his wife Michelle said in a statement thatFranklin, who performed at his first inauguration in 2009, "helped define the American experience" through her "compositions and unmatched musicianship".
"In her voice, we could feel our history, all of it and in every shade - our power and our pain, our darkness and our light, our quest for redemption and our hard-won respect," the Obamas added.
"She helped us feel more connected to each other, more hopeful, more human. And sometimes she helped us just forget about everything else and dance.
"Aretha may have passed on to a better place, but the gift of her music remains to inspire us all. May the Queen of Soul rest in eternal peace. Michelle and I send our prayers and warmest sympathies to her family and all those moved by her song."
President Donald Trump said that Franklin "was a great woman, with a wonderful gift from God".
Dozens of stars across the worlds of music and entertainment paid tribute to Franklin,including Mariah Carey, who thanked her as "my inspiration, my mentor and my friend".
She tweeted: "Aretha Franklin. The Queen of Soul. The Icon. The ultimate singers' singer. The greatest singer and musician of my lifetime. The power of your voice in music and in civil rights blew open the door for me and so many others."
In a second tweet, Carey wrote: "You showed me I could sing the songs I wanted to sing and bring God with me. You've inspired millions everywhere yet never left home, never left church. Today the entire world mourns your home-going and celebrates all the brilliance you left behind."
She added in a third: "I will forever cherish the moments I spent in your presence. Your indelible impact was earned not only by your incomparable voice but by your bigger than life personality, wit and humor.
"I say a BIG prayer for you. You will forever have all our RESPECT. Love, Mariah."
American TV star and actress Oprah Winfrey simply wrote: "We'll miss you Queen. #ArethaQueenForever."
Music star Adele said she was "absolutely heartbroken" by Franklin's death.
Posting on Instagram, she said: "I can't remember a day of my life without Aretha Franklin's voice and music filling up my heart with so much joy and sadness. Absolutely heartbroken she's gone, what a woman. Thank you for everything, the melodies and the movements."
Sir Elton John, who was present for Franklin's last ever performance, called her death a "blow for everybody who loves real music: Music from the heart, the soul and the Church".
He added, on Instagram: "Her voice was unique, her piano playing underrated, she was one of my favourite pianists. I was fortunate enough to spend time with her and witness her last performance - a benefit for the Elton John Aids Foundation at St John The Divine Cathedral.
"She was obviously unwell, and I wasn't sure she could perform. But Aretha did and she raised the roof. She sang and played magnificently, and we all wept. We were witnessing the greatest soul artist of all time. I adored her and worshipped her talent. God bless her."
Sir Elton's husband David Furnish said that her final performance is one he will never forget.
He added on Instagram: "Elton and I went to greet her before the show and were taken aback at how thin and frail she appeared. She looked at Elton with the sweetest smile and said 'I didn't want to let you down'. Such class. The performance she generously gave everyone was overwhelmingly moving."
Furnish said Franklin had channelled the "spirit of God" in the church, and that his husband "couldn't stop crying".
Her friend, record producer Quincy Jones, wrote on Twitter: "From the time that Dinah Washington 1st told me that Aretha was the 'next one' when she was 12-years old, until the present day, Aretha Franklin set the bar & she did it with the professionalism, class, grace, & humility that only a true Queen could...
"I treasured every moment that we spent together from working in the recording studio, to performing on the steps of the Lincoln Memorial, or simply hangin' in the kitchen, & I will miss her dearly. RIP Ree-Ree... You will reign as the Queen forever."
Beatles star Sir Paul McCartney asked for fans to take a moment to "give thanks to the beautiful life of Aretha Franklin, the Queen of our souls, who inspired us".
He added, in a tweet: "She will be missed but the memory of her greatness as a musician and a fine human being will live with us forever. Love Paul."
Barbra Streisand shared a picture of herself with Franklin taken six years ago, writing on Instagram: "This photo was taken in 2012 when Aretha and I performed at a tribute celebration for our friend Marvin Hamlisch.
"It's difficult to conceive of a world without her.
"Not only was she a uniquely brilliant singer, but her commitment to civil rights made an indelible impact on the world."
Diana Ross tweeted: "I'm sitting in prayer for the wonderful golden spirit ArethaFranklin."
Annie Lennox wrote: "As the One and Only 'Queen of Soul' Aretha Franklin was simply peerless. She has reigned supreme. And will always be held in the highest firmament of stars as the most exceptional vocalist, performer and recording artist the world has ever been privileged to witness.
"Superlatives are often used to describe astonishing artists, but in my view even superlatives seem insufficient. Everyone who loved Aretha will be saying little prayers of gratitude, appreciation and respect for the musical life force that enriched our lives.
"Her voice will soar on forever..."
Franklin duetted with Lennox's band the Eurythmics on the 1985 hit Sisters Are Doin' It For Themselves.
Sir Mick Jagger said in a statement: "Very sad to hear the news about Aretha, she was so inspiring and wherever you were she always brought you to church."
Sir Ringo Starr tweeted: "God bless Aretha Franklin the queen of soul and peace and love to her family."
Other stars from across all genres of the music world, including Paloma Faith, Liam Gallagher and Bette Midler, praised Franklin for her powerful voice following her death.
The official Twitter account for Nasa also paid tribute to Franklin.
"We're saddened by the loss of Aretha Franklin. Asteroid 249516 Aretha, found by our NEOWISE mission and named after the singer to commemorate the #QueenOfSoul, will keep orbiting beyond Mars," the tweet read.
Franklin had the asteroid named in her honour in 2014.
Labour leader Jeremy Corbyn tweeted that "we've lost one of the world's most wonderful singers" and praised her music as "moving and full of hope".
Former American secretary of state Madeleine Albright tweeted: "Aretha Franklin was the Queen of Soul, and she was also an incredible ambassador for American values and culture. In every corner of the globe, Aretha's voice brought people joy and garnered more R-E-S-P-E-C-T for our country. I was honored to know her and will miss her dearly."
BBC Radio 2 announced a change in its schedule following Franklin's death.
"Following the sad news of the death of music legend Aretha Franklin, you'll be able to hear a special tribute to The Queen Of Soul presented by Paul Gambaccini, tonight at 9, here on BBC Radio 2," a statement said.