Probably. The course of the world would look much different today, in some shape or form.
And it would have been BLOODY.
I once read an analysis that tried to guess at how things would have gone had either side decided to go to the mats in 1945/46.
Your suggestion that it would have been BLOODY is spot on. Both sides had advantages that it would have been extraordinarily difficult for the other to overcome.
The Soviets had a substantially larger and more powerful land army. They had more experience in land warfare and a better main tank.
The United States (even before including Britain) was vastly superior both quantitatively and qualitatively in both naval and air forces. Additionally, the United States had a much better logistical capability and a LOT more oil.
The conclusion was that a quick victory would have been unlikely for either side. The attacker would have likely had significant initial success based simply on the advantage of surprise but once that faded the two sides would likely have been dragged into a bloody stalemate.
If the Russians attacked first, they'd have had great initial success with their larger army and huge numbers of T34's. Their air force, however, would have been overmatched by the B24's, B17's, B29's, Lancasters, and Halifaxes (last two assumes that Britain joins in) and while they certainly would have shot some US Bombers down, they had virtually no Strategic Air Force of their own with which to retaliate.
Furthermore, the immediate end of American Lend Lease aid would have left the Soviets with finite quantities of certain critical supplies particularly jeeps, trucks, and locomotives. This would have created a logistical nightmare that would have been made infinitely worse by their lack of spare parts for existing US-built equipment and by US Strategic air attacks on EXTREMELY long Soviet supply lines.
The Soviets would also have had to contend with a blockade that the US and Britain would obviously have instituted and the fact that with massive numbers of carriers and carrier aircraft available the US and Britain could have launched attacks anywhere along any coast and within a few hundred miles of shore with almost no warning. This would have forced the Soviets to try to defend an enormous amount of airspace that they simply wouldn't have had enough planes to defend leaving them vulnerable to random attacks basically anywhere the US happened to decide to pounce.
US Naval Supremecy and Air Superiority would have given the US a freedom of maneuver which would have made up for much of the USSR's numerical superiority because the US would have had a much greater ability to concentrate forces.
If the US had attacked first they would have had initial success with their superior air force but as you move East from western to Central to Eastern Europe the continent gets much larger North-to-South. At roughly the Franco-German border it is only about 600 miles from the North Sea to the Mediterranean but the distance from Sevastopol to Murmansk is many times that great. As the front line gets longer, the impact of air power is necessarily diluted. It becomes more of a numbers game and . . . The Soviets had the numbers.
Additionally, as both Napoleon and Hitler learned, Russia is really, Really, REALLY big.
It isn't even necessarily clear that the Atomic Bomb would have ended the war. When the US dropped the bombs on Hiroshima and Nagasaki the Japanese had almost no surviving airforce, less experienced pilots, and still less fuel. Furthermore, Japan is an island nation that by that time functionally had no navy so the B29's spent very little time over enemy territory. Bombing Moscow from France or Britain is an entirely different operation and you run the risk that a shot down aircraft may result in Soviet capture of an intact bomb and potential Soviet reverse engineering of same.
Ultimately I think that the US would have prevailed simply because in 1945 US GDP was roughly 50% of Global GDP and several times greater than Soviet GDP. This would have given the US the ability to build what they needed to win but it would have been insanely bloody.