The claim that Russia is not directly responsible for the sheer amount of destruction in Aleppo, or really any Syrian city, is technically true. Syria's cities have been pounded by air force, artillery, tanks, mortars, and small-arms fire for years, in many cases dating back to the early spring of 2012, and in some cases before that. Aleppo itself has been subjected to open warfare since the summer of 2012, more than four years ago.
However, this technicality ignores an important point -- that Russia has supplied the Assad regime with a substantial amount of bombs, munitions, and aircraft parts, much of which has been delivered since Assad's bombing campaign began. Then Russia began bombing Syria's cities directly. Furthermore, while this was going on Russia used its veto power on five occasions to stop the United Nations Security Council from taking actions to address the conflict.
In other words, even if Russia is not directly responsible for most of the destruction of Aleppo, it has played a key role in the city's devastation, and the widespread damage to Syria's infrastructure and its civilian populace.
Before the worst parts of the Assad government’s fight against its protesting population began in February 2012, Russia began supplying the Assad regime with ammunition, training aircraft, anti-tank missiles, air defense systems, as well as helicopters and equipment needed to maintain Syria's air force. At that point the UN was already warning that 70,000 people had been killed in the crisis. By 2014, Russia bolstered its efforts to provide the Assad regime with direct military assistance. Reuters reported that increased arms supplies from Moscow to Damascus included armored vehicles, drones, guided bombs, surveillance equipment, radar and electronic warfare systems, spare parts for military aircraft and tanks, and other key supplies.
In August 2015, a month before Russia's air campaign began in Syria, Human Rights Watch released a report on the crisis that had a very clear conclusion: that the government of Bashar al-Assad's use of barrel bombs was "the greatest threat to Syrians." The improvised devices were being dropped out of Assad's helicopters -- the ones maintained by Russia -- onto cities below. The improvised weapon is indiscriminate and most effective against soft targets -- civilians. Furthermore, that weapon was being increasingly relied upon because the Syrian air force, not because it was depleting its inventory of traditional Russian-manufactured bombs -- it still uses those -- but because they're terrifying to the population below. The Syrian Network for Human Rights (SNHR) released a report in February 2016 that concluded that Russia had killed more civilians in the preceding month than either the Syrian military or the extremist group Islamic State (IS) combined. In September 2016, a little more than a year after Russia began its air campaign in Syria, the UN Commission of Inquiry concluded that the Syrian and Russian air forces were causing the greatest number of casualties in the country.
These are just a few of the reports released by international observers which reach similar conclusions, and while the focus of such reports is typically on the damage to human beings, rather than buildings, it follows that there is a correlation.
So whether the devastation in Aleppo and elsewhere in Syria is the result of munitions dropped by the Russian air force, or by the Assad regime that has been directly supplied by the Russian military, the result is the same -- Assad and those who are fighting for him are responsible for the majority of the damage in Syria. Since groups like the UN have been reporting on the Syrian military's role in killing so many since the start of the conflict, Russia is ultimately responsible for providing the murder weapons even when it is not pulling the trigger.
It's also worth noting that while this was going on, Russia has used its veto power five times to stop the United Nations from addressing this crisis, effectively ensuring that no international body take it upon itself to stop Assad from destroying the country.