Bush rejected any easing up of the US sanctions without a full transition to democracy and said doing so would only bolster the communist government's grip on power.
Castro, 81, has not been seen in public in 15 months because of an undisclosed intestinal illness.
Many analysts believe that a stable transfer of power has already taken place and that Raul Castro will remain in charge of Cuba's government when Fidel dies.
But Bush, reinforcing the administration's hardline policy towards Havana, said such a transition amounted to merely "exchanging one dictator for another".
"America will have no part in giving oxygen to a criminal regime victimising its own people," Bush said in a speech at the state department on Wednesday, where he appeared with family members of Cuban dissidents.
"We will not support the old way with new faces, the old system held together by new chains.
"You have the power to shape your own destiny."
But he also made a direct appeal to the Cuban military and security forces not to stand in the way if citizens push for democratic change.
"When Cubans rise up to demand their liberty, the liberty they deserve, you've got to make a choice: Will you defend a disgraced and dying order by using force against your own people, or will you embrace your people's desire for change?"