Yes and the constitution was written this way to ensure this was the case.
Article I Section 8:
The Congress shall have power [...]
To declare war, grant letters of marque and reprisal, and make rules
concerning captures on land and water;
Article II Section 2:
The President shall be commander in chief of the Army and Navy of the
These powers were intentionally separated with the intent of preventing any one branch of government from being able to wage war. The intent was that the president needed the permission of congress to enact military action, and that congress would need the president to act and actually wage war.
However after WWII and the formation of NATO the president has utilized the vote ratifying NATO as express consent to fulfill our military obligations of the treaty to project our military presence beyond our borders. And its approval of our joining the UN as permission to use military force at its bequest.
As a result Truman used UN resolution 83 to deploy ground forces in Korea with out congressional support, and by calling it a police action. A similar reasoning was used in Vietnam also calling it a police action. All of this was allowed because of the advent of the Nuclear war:
Nuclear war was seen as the most realistic war-fighting scenario, with
all other forms of war trivial in comparison. Just as nuclear weapons
came to be called “strategic weapons” with other weapons of war
occupying a lesser space, nuclear war became identical with war in
general. If that was so, then constitutional procedures that could not
be applied to nuclear war were simply no longer relevant.
Paradoxically, if nuclear warfare represented the highest level of
warfare, there developed at the lowest level covert operations. Apart
from the nuclear confrontation with the Soviets, there was an intense
covert war, from back alleys in Europe to the Congo, Indochina to
Latin America. Indeed, it was waged everywhere precisely because the
threat of nuclear war was so terrible: Covert warfare became a prudent
alternative. All of these operations had to be deniable. An attempt to
assassinate a Soviet agent or raise a secret army to face a Soviet
secret army could not be validated with a declaration of war. The Cold
War was a series of interconnected but discrete operations, fought
with secret forces whose very principle was deniability. How could
declarations of war be expected in operations so small in size that
had to be kept secret from Congress anyway? SOURCE
The result was The War Powers Act after the enormously unpopular Vietnam war that would limit the amount of time a president could deploy troops with out congressional approval.
Many believe this act has had the effect of increasing the powers of the president potentially making this unconstitutional. But there is no law that permits Congress to deploy troops on their own. That responsibility lies solely in the Commander in Chief.
There is recourse for Congress should a president decide not to wage a war that Congress declared. But these recourse's also exist for any action Congress deems inappropriate. The most severe being Impeachment, with probably the most effectively meaningless being censure.