Villas-Boas made Damiao his top January transfer target but, despite a last-ditch attempt, Spurs failed to land the 23-year-old before Thursday night's 11pm deadline.
Chairman Daniel Levy's failure to sign Damiao means Jermain Defoe and Emmanuel Adebayor will be Tottenham's only recognised senior strikers for the rest of the campaign, and the latter may not be available for up to another two weeks as he is away at the African Nations Cup.
Internacional claimed they did not let Damiao leave for White Hart Lane because Spurs' offer, reported to be £13 million, did not meet their valuation of a player who has 14 caps for Brazil and was top scorer at the Olympics.
Villas-Boas admits being left with two strikers is a "risk", but brushed off suggestions the club should have pushed the boat out to sign Damiao.
"We've approached this market very sensibly," the Tottenham manager told a press conference.
"I've said in our conversations that if (the Damiao bid) was going to go out of proportion, we would step out of it.
"We have made it public before that he is a player that we chase, but we couldn't agree on the full conditions so we've moved on."
Tottenham, who have a club partnership with Internacional, were expected to renew their interest in Damiao this summer, but Villas-Boas claims that is now unlikely.
"No I don't think (it will happen in the summer)," Villas-Boas said. "It's probably finished now."