The US president has instead replaced it with a softer approach stressing "new partnerships" and multilateral diplomacy.
"Our long-term security will not come from our ability to instill fear in other peoples but through our capacity to speak to their hopes," Mr Obama said in a message introducing a new national security strategy.
In the 52-page document, drawn up after 16 months of deliberations, Mr Obama outlines a much broader set of priorities and methods than Mr Bush's tightly-focused determination to eradicate Islamism by any means possible and alone if necessary.
"We will always seek to delegitimise the use of terrorism and to isolate those who carry it out," it states. "Yet this is not a global war against a tactic – terrorism – or a religion – Islam.
"We are at war with a specific network, al-Qaeda, and its terrorist affiliates who support efforts to attack the United States, our allies, and partners."
Mr Obama distances himself from Mr Bush's concept of pre-emptive wars to prevent emerging threats, instead citing the national security implications of global economic crises and climate change.
American global leadership, the document argues, depends on a strong economy and a determination to progress in the areas of "education, clean energy, science and technology, and a reduced federal deficit".
It highlights home-grown terrorists who become "radicalised" on American soil. "Our best defences against this threat are well informed and equipped families, local communities, and institutions.
"The Federal Government will invest in intelligence to understand this threat and expand community engagement and development programs to empower local communities."
It does note that for more than a decade, the United States has been involved in a struggle against a "far-reaching network of violence and hatred". Military superiority will remain "a cornerstone of our national defence and an anchor of global security".
But also important will be "new partnerships with emerging centres of influence" and a "push for institutions that are more capable of responding to the challenges of our times". American innovation is "a leading source of American power".
The document presents the Obama administration as realist in nature, an implicit rebuke to the neoconservatives who hoped to reorder the world based on American values. "To succeed, we must face the world as it is," it states.
There should be tough engagement "without illusion" with foes like Iran and North Korea but isolation would be the result of their continued intransigence. In his last national security strategy in 2006, Mr Bush declared that "the war on terror is not over".
In a preview of the document, John Brennan, Mr Obama's senior counter-terrorism adviser, said that there was a "new phase" in al-Qaeda tactics in which terrorists who did not fit the "traditional profile" would carry out attacks.
These included Umar Farouk Abdulmutallab, the Nigerian who attempted to explode an underpants bombs on a Detroit-bound plane on Christmas Day, and Faisal Shahzad, the Pakistani American accused of leaving a car bomb in New York's in Times Square this month.
"As our enemy adapts and evolves their tactics, so must we constantly adapt and evolve ours, not in a mad rush driven by fear, but in a thoughtful and reasoned way that enhances our security and further delegitimises the actions of our enemy," Mr Brennan said.
It was wrong, he added, to "describe our enemy as jihadists or Islamists" because that would "play into the false perception" that al-Qaeda and its allies were "religious leaders and defending a holy cause, when in fact, they are nothing more than murderers".
