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Alcaraz came from behind to beat Jannik Sinner before Medvedev edged out American Tommy Paul 1-6, 7-6(3), 6-2 in the second semifinal.
Sakkari got the better of US Open champion Coco Gauff in a hard-fought second semifinal.
Third seed Coco Gauff served up 17 double faults but still managed to beat China's Yuan Yue 6-4, 6-3 to reach the women's semifinals in the Californian desert for the first time.
Navarro, seeded 23rd, broke the hard-hitting Belarusian for a fourth time on match point to book a quarterfinal clash with ninth seed Maria Sakkari of Greece.
The world No. 1 is seeking an unprecedented sixth title at the Masters 1000 event that begins next week and is regarded as the biggest tennis tournament outside of the four Grand Slams.
The world No. 2 will not play at the Indian Wells tournament.
Sania is the nominee from the Asia/Oceania Zone along with Indonesia's Priska Madelyn Nugroho.
The shoulder problem had also forced the 31-year-old to withdraw from her second-round match in St Petersburg, Russia, last month.
Results 1-8
