From a new restriction on abortion to trying to limit marriage to the union between a man and a woman, some of the White House measures are certainly welcomed by Christian conservatives in the US.
But in many cases, other parts of the government and Mr Bush's party have been promoting the agenda with the chief executive only coming into the frame much later, at least in the public eye.
This may well change in the upcoming election year, and the Democrats will be launching their own claims to the moral high ground as they try to win back the White House.
Faith is a key part of American society in general, with about 60% of citizens saying religion is "very important" in their lives according to a Gallup poll in November.
The way senior politicians talk so freely about their own faiths can seem strange in other countries - Gallup reports just 28% of Canadians and 17% of Britons say religion is "very important" to them - but it is commonplace in the US.
And with religion being so central to so many people - another poll by the Pew research group in November found 81% of Americans saying that prayer was an important part of their daily lives - issues connected to faith will feature in the election.
Faith in the system
The Pew survey indicated that in the late 1980s, the majority Protestants were split fairly evenly between Republicans and Democrats. But now nearly twice as many say they are Republican as Democrat.
White Roman Catholics, once solidly Democrat, are now divided almost evenly between the parties, the survey suggested.
Christian conservatives have become a base of the Republican party and will continue to be courted by the Bush team with various observers suggesting that the 2004 presidential poll will see the parties concentrating more on energising their core of supporters to go out and vote rather than seeking backing from independents.
But the Democrats are not going to give up on those votes and their candidates are at pains to show that they belong on the moral high ground too as they hope to retake the White House.
With more than a nod to the importance of religion in American life, Democrat frontrunner Howard Dean, who had hitherto run a largely secular campaign, recently decided it was time to "get into a little religion".
"I think religion is important and spiritual values are very important, which is what this election is really about," he told voters in Iowa in December.
Other Democrat candidates have also been keen to emphasise their values.
Dick Gephardt stressed ethics when he talked to voters about corporate greed, saying: "We've lost a sense of right and wrong," and he often says the current healthcare system is immoral.
Faith is an important part of Joe Lieberman's life and is often brought up, not least because he would be the first Jewish candidate to stand for the White House from either main party.
But he also makes a broader point, relevant to many Democrats. "It angers me that Republicans seem to suggest they have a monopoly on values in public life," Mr Lieberman told the New York Times.
"They don't. We don't either, but we think about values, we care about values including faith-based values."
Moral morass
American voters certainly want to know what prospective leaders think about moral issues, but two high-profile issues - abortion and gay marriage - could cause problems for Democrats.
On abortion, the Democrat base will not countenance weakening of the historic Roe vs Wade judgement which legalised termination of pregnancies.
A ban on late-term abortion signed recently by President Bush would have been supported by 68% of Americans, an earlier Gallup poll suggested, but all the Democratic candidates opposed it.
Even more divisive for Democrats could be the matter of gay marriage. In the summer, Mr Bush said society should respect all individuals though he said even then he believed marriage should only be between a man and a woman.
But after the Massachusetts Supreme Judicial Court said marriage between homosexuals should have the same legal standing as traditional unions, his words became much stronger, saying the judgement "violated" an important principle.
"I will work with congressional leaders and others to do what is legally necessary to defend the sanctity of marriage," he asserted.
With that, the president again reflected the views of the majority of Americans. A December Gallup poll found 65% of respondents opposed to the recognition of gay marriage, with half of those saying they felt strongly on the subject.
The liberal heart of the Democratic Party would be opposed to restricting the rights of homosexuals and it may be something which Republicans could use if Howard Dean becomes the official challenger - for as Vermont governor he signed a controversial bill legalising civil unions for gays within his state.
Congress role
But while moral matters may end up being seen as vote-winners, the White House has so far let other members of the party, such as Pennsylvania Senator Rick Santorum, lead the fight while Mr Bush mostly stays out of the fray until later.
The president may be passionate on the subjects - an end-of-year release about his achievements called late-term abortion an "abhorrent practice... that offends human dignity" - but he left the arguments on that to Congress, and it is Congress he is expecting to address gay marriage.
One reason for a lower profile could be that in what surveys suggest is an increasingly polarised society, Mr Bush does not want to give potential supporters a reason not to vote for him.
And while jobs, the broader economy, healthcare and other subjects are set to play larger roles, it would be wrong to discount issues of more moral or religious importance from the election.
In the 2000 race, the choice for president of five states was decided by less than 1% of the vote. If there is a close contest again, a simple question of morals could tip the balance.
