Mr Bush joins several high-profile Republicans who have refused to support the New York businessman's campaign.
House Speaker Paul Ryan said on Thursday he "was not ready" to support Mr Trump, but will meet him next week.
Breaking with tradition, Mr Bush's father and brother - both former presidents - also withheld support.
Some Republicans have said they would back Democrat Hillary Clinton but Mr Bush ruled that out.
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"Donald Trump has not demonstrated that temperament or strength of character," Mr Bush said. "And, he is not a consistent conservative. These are all reasons why I cannot support his candidacy."
Supporting:
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Yet to comment:
Mr Bush had previously pledged to support the eventual Republican nominee while he was still a candidate for president.
Republican Senator Lindsay Graham of South Carolina also announced on Friday that he would not vote for Mr Trump.
"I think Donald Trump is going to places where very few people have gone and I'm not going with him," Mr Graham told CNN.
The Trump campaign swiftly responded to Senator Graham who also was a Republican candidate for president.
"While I will unify the party, Lindsey Graham has shown himself to be beyond rehabilitation. And like the voters who rejected him, so will I," Mr Trump said.
Many Republican candidates for lower offices are concerned about running on the same ballot as Donald Trump, who has alienated minority voters through his rhetoric about building a wall with Mexico and banning US entry to Muslim travellers.
Many Americans choose to vote for either the Democrat or Republican Party, rather than weighing the individual candidates.
Republican representatives fear that voters who oppose Trump may eschew the Republican Party all together.
Some Republicans, including a former top adviser and speechwriter to Senator McCain, have begun to openly call for the party to oppose the presumptive nominee and to work to independently elect a conservative candidate, such as Nebraska Senator Ben Sasse, who has indicated that he will not be supporting Mr Trump.
Former Florida Governor Jeb Bush has said he will not vote for Republican nominee Donald Trump in November's presidential election.