Thermodynamically No
<PERSON><PERSON> gives a good answer about aerosol effects. Others observe that impacts add energy to the planet. I would simply like to look at the energy budget.
The sun dumps on the order of 173 petawatts on the earth, nonstop. Now, asteroids have a heat capacity on the order of $10^{18} - 10^{24} J/K$, depending on size. Note that a petawatt is $10^{15} J/s$. Also note that the sun dumps $10^{18} W$ on the planet every 5.8 s or so.
Now, let's imagine we find an asteroid floating around interstellar space at a chilly 3 K (or colder or warmer...the exact number doesn't matter). Let's further suppose we can chuck it at the earth and land it without any significant addition of energy (we use an alien tractor beam to gently lower it to the surface). This asteroid will "cool" the earth by heating itself up. So...how much cooling should we expect?
Instead of pretending that the earth is a lone ice box floating in space, and calculating what happens when we drop a large ice cube into it, I think it's more useful to simply ask: "How quickly will the sun melt our ice cube?"
The sun will contribute enough energy to warm the asteroid by 1 K every 6s or so, for a small asteroid. To raise the asteroid from 3 K to earth mean surface temperature of 288 K will take somewhat less than 1800 s, or about 30 minutes. 912
That's about how long an ice cube would last in a chilled beverage on a not particularly warm day.
Now, that is roughly equivalent to turning off the sun for 30 minutes. That would have a noticeable effect on global weather, but I doubt it would remain noticeable even a few days after. Over the course of a year, this blip would amount to a solar energy reduction of 0.005%. Hardly enough to detect in the climate record.
Of course, we picked the smallest asteroid for the initial calculation. If you go towards the large end, you can crank up all these numbers by a factor of a million, more or less. Ceres, for instance, would suck up about 5000% of the sun's annual energy sent to earth. That's 50 years of solar energy, and obviously a significant impact. Given that it's round, we can debate whether Ceres counts as an "asteroid" or not. Nysa is more "asteroid-shaped", but only has about 100x the heat capacity of our initial calculations. So, a solar dip of 0.5% over the course of a year might be enough to notice, but likely not enough to cause attributable long-term climate effects (given that the <PERSON> cycles produce larger variance).
Note that if you start with a more typical asteroid temperature of about 200 K, you would reduce all the energy budget numbers to about 1/3 of their value (meaning, the smallest ice cube would melt in ~10 minutes). 912