As a practical matter, they can't. But in theory, yes they could, and a veto of a budget-busting spending bill is an example of one that could be vetoed, putting the heat on Congress to take all the blame for the deficit if they choose to override the veto. Most congress critters would be very nervious about doing that.
As to the start of the conversation on Reagan's deficit spending, lets be honest. Reagan allowed it to happen. He could have easily stopped it if he was of a mind to.
Despite the doubling of Federal revenues over 8 years as a result of his 1982 tax cuts, Reagan and Tip O'Neill came to an agreement. Reagan wanted to increase military spending to bankrupt the Kremlin. O'Neill said he would go along as long as he had a free hand in domestic spending. Reagan agreed seeing his first priority as defeating the Soviets and ending the Cold War before it got hot. He allowed the democrats in Congress to run crazy with the national credit card.
Was it the right decision? I guess it depends on how much it was worth to see the Berlin Wall come down and for half a billion people to move from slavery to freedom.
When a President has a hostile Congress he can veto some but they will mark him and make him pay, crushing any initiative he tries to get through. Reagan had to pay the piper, but he won in the end and his legacy is defeating the evil empire.
We must put the blame for the mess in its proper place and when it comes to spending, tis Congress that do the dirty deed.
I give Presidents their due for what they can, wars and treaties and the most important of all reaffirming that America is good, the shinning city on the hill. Reagan did that and more. Few since have even come close.
Mitt and Huck are RINOs when it comes to such. BOTH good men but not good enough. The closest I have seen to match Reagan (so far) is that SuperMom from Alaska and she has an uphill battle to win that right to run. I wish her well.
I like Fred Thompson because he has that Reagan panache but he does not seem to "want" it bad enough to stake his claim. Time will tell.