As is often the case, no in theory, yes in practice. The president’s responsibility as to the effective date of laws is that he either signs them or doesn’t. However, most legislation of significant effect delegates power to the Executive Branch agencies for implementation. It’s not uncommon for a law to say a certain provision will go into effect on X date, unless the Secretary of Whatever certifies that, for reasons authorized in the statute, it should be delayed. Alternatively, you can have a law which operates in one way at the start but, if the problem at issue gets worse, other, more aggressive measures kick in. (This is one of the proposals in the current health care debate – the so-called “trigger” option. This being GQ, I shall refrain from saying how stupid it is. Whoops!) And those would typically be initiated by a (probably statutorily mandated) report from an agency.
In some cases the discretion given to the agency is extremely limited – such as a requirement that you measure, oh, let’s say air quality and, if it’s above X, you must immediately do Y. Other times it’d be much broader, for instance a statutory mandate that the military do this and this unless “in his discretion, the Secretary of Defense determines that it would have a deleterious effect on national security.”
So to headline it – the effective date of a statute is specified in the statute (or, if not, it’s effective immediately upon enactment). But the functional effective date of particular substantive provisions of the statute can be conditioned on agency action. And while the specified agency actions can sometimes be merely ministerial, in other cases they can be exercises of significant discretion.
(This discussion assumes that the agency heads will do what the president tells them to do, which is roughly true most of the time, but precisely true none of the time.)