Pique, who plays for treble-winning Catalan club Barcelona, angered Spain supporters who are also fans of Real Madrid by mocking Real forward Cristiano Ronaldo after Barca won the Champions League final last weekend.
Pique was whistled before and during Thursday's 2-1 friendly win over Costa Rica, prompting Del Bosque to leap to his defence and scold local media for what he said was fuelling "violence and hatred" between Spanish regions.
"I don't think what happened is right," Del Bosque told a news conference.
"Whistling one player is the same as whistling the whole team," he added.
"It leaves me with a very bad taste. The most important thing is that we are united."
Pique's treatment has its origins in the historic tension between Madrid, which represents central government, and Barcelona, where many are strongly in favour of independence from Spain.
Along with many Barca players who are also proud Catalans, like Xavi, Carles Puyol, Cesc Fabregas and Sergio Busquets, Pique has been a key part of Spain's recent success, helping them to a debut World Cup triumph in South Africa in 2010 and a second straight European Championship two years later.
They will be bidding for a third consecutive continental title at the tournament in France next year.
With five matches played, Spain are second in qualification Group C on 12 points ahead of their trip to Borisov, three behind leaders Slovakia, who beat them 2-1 in Zilina in October.
Ukraine are third on nine points and Belarus fourth on four, with Macedonia (3) and Luxembourg (1) in fifth and sixth.
The top two teams qualify directly, as well as the best third-placed side.
Belarus coach Alexander Khatskevich admits Spain are far superior but is hoping hard work will help the eastern Europeans exploit some of their weaknesses.
"I will accept any result as long as there are no half-hearted players in the team," Khatskevich was quoted as saying on UEFA.com on Friday.
"Of course Spain are better than us in every aspect but we can fight them with our team game and try to exploit their weaknesses," he added. "And they have those weaknesses, trust me."
(Additional reporting by Dmitriy Rogovitskiy, editing by Ed Osmond)
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