Top seed Nadal, an 11-time champion in Barcelona, eased to a 6-3, 6-2 win over world number 13 Carreno Busta in 90 minutes.
Fognini denied wrongdoing and was upset as he left the court, breaking his racket after hitting it at the net post on his way out.
Roberto Bautista Agut defeated fellow Pablo Andújar 6-4, 6-0 to advance to the round of 16 in the Barcelona Open.
Sumit Nagal overcame a tough challenge from Illya Marchenko in the ATP 500 Barcelona Open qualifier to be one step away from the main draw of the tournament.
Nadal extended his dominance on the surface with a 6-2, 6-1 win over 19-year-old Greek Stefanos Tsitsipas on Sunday to win his 11th Barcelona Open title.
The Serbian was beaten in three sets by Austrian Dominic Thiem in the Monte Carlo Masters third round, but said he was pain-free for the first time in two years.
Nadal yesterday defeated compatriot Albert Ramos-Vinolas 6-1, 6-3 to clinch his 10th singles title of the Monte Carlo Masters.