Stalin did indeed attack Finland during the Summer of '44, with the Soviet Karelian offensive starting less than a week after the Allies landed in Normandy. The goals of this offensive are somewhat debated, but in general, the goal of the offensive was to force Finland out of the war. It's possible that Stalin was willing to annex Finland completely should the offensive be a massive success and the Finnish forces completely broken, but this was, at most, a "best case" scenario, and this would have probably raised the ire of the western Allies. In general, Finland managed to maintain its independence through western sympathy, political negotiations in the Tehran conference between the allied countries, dogged resistance, and geopolitics.

During the Tehran conference in 1943 the Allied leaders discussed the war, and the post-war future of Europe. The British and the Americans wanted to stop the Soviets from annexing the whole of eastern Europe (in the end failing miserably) and demanded, for example, free elections in the Baltic states on joining the Soviet Union. This did not come to pass. However, with these decisions and talks on the future of Europe, the question of Finland was also on the table. The winter war had endeared Finland to the western public, the country having been attacked by the Soviet Union without provocation, and having lost 10% of its territory to the superpower, even the country joining Barbarossa was seen as somewhat understandable. Despite the official state of war between Britain and Finland, the countries kept unofficial diplomatic relations through Stockholm and the Swedes throughout the war.

This endearment was also shown in Tehran. Churchill told Stalin that despite understanding the Soviet security needs around Leningrad he hoped the Soviets would not annex Finland, and Stalin told that he believed in independent Finland, although insisted on heavy reparations. Of course, it's hard to tell if this "independent Finland" at this stage was similar to the independent Poland, Romania, and Hungary that were to follow the second world war. However Stalin also famously stated that "any such country that has fought for its independence as bravely as Finland has, deserves respect". In hindsight this statement is almost prophetic.

At this point I think we have to discuss the geopolitical position of Finland. Situated on the northern Baltic coast, the country is an important choke point on the bay of Finland and for controlling the Baltic. However, in terms of war with Germany Finland was an additional direction of operations. Unlike Poland, Romania and Hungary, which acted as major attack routes towards the heart of Germany, Finland was a sidepath that led nowhere. With 3 heavy years of war even the Soviet war machine was starting to show some signs of exhaustion, its manpower reserves were not inexhaustible and the forces used on the Finnish front could be used somewhere else. Thus having Finland out of the war would simplify the Soviet war effort, especially if the Soviets could get the Finns to drive the remaining German forces out themselves.

So, with the Tehran conference done, Soviet Union started negotiating demands for peace. The Soviet demands were however rejected by the Finnish government. The terms were based on the Moscow peace treaty, and Finland would've had to give up Petsamo, pay 600 million dollars of reparations (9 billion in today's money, with a much weaker Finnish economy) , demobilize the army, cut ties to Germany and expel or imprison the German troops in Finland. Even though Finland itself had done some forays into peace negotiations earlier, the terms were simply seen as too harsh, and Finland feared that the Soviets would just annex the country after the peace.

Much has been written about the offensive in other places, so I shall not go into detail here. In short despite initial Soviet success and the capture of Viipuri on the initial timetable, on the 20th of June, the attack soon stalled due to heavy resistance from the Finns who had also received important aid from Germany: new armoured vehicles and the flight group Kuhlmey. The Soviet demands for surrender helped bolster the Finnish morale, as the Finns feared the fate of the Baltic or Poland. During the heavy fighting in Tali-Ihantala, the Finns did something that even the Germans didn't manage: stopped a full blown Soviet offensive. The Soviet forces were spent and blocked from advancing towards the finnish heartland and the capital, and soon started moving troops from Finland towards the German front. However the finnish forces were exhausted, with little replacements to be had and almost all of the ground gained during the first year of Barbarossa was lost. The Finns had managed the impossible, but knew that they could not do it a second time. Despite, the sun was definitely setting on Hitlers Germany with the massive success of Bagration. However the quick victory and occupation of Finland was also proven impossible, and with changing Finnish attitude towards the war Stalin couldn't simply try and annex the country by force of arms without blatantly breaching the things agreed upon in Tehran.

Thus both sides came to the table. The Soviets dropped the word "surrender" wholly from their vocabulary regarding Finland, whilst the Finns pretty much agreed to the post-Tehran demands with some additional ones made. The Finns would later expel the German forces from its borders during the Lapland war.

Unlike Romania, Hungary and Poland, Finland had maintained its independence and (relative) political sovereignty through great power politics, being the plucky underdogs, the relative unimportance of its geopolitical place and, most Finnish of all, sheer determination and being too damn stupid to give up.

However even if Finland managed to keep its independence, the threat was far from gone. In the coming years, Finland lived in fear of a communist coup and Soviet annexation to the point that the time period is known as the "years of danger" that wouldn't end until 1948 when Finnish and Soviet relations were finally normalized. Finland would get its own dictator-lite, and a little thing called finlandization happened.

But that's a whole other story.