The Soviet leadership greatly underestimated the importance of the Watergate affair and did not understand its true political significance. When former ambassador Averell Harriman visited Moscow in 1976, he reported that General Secretary Brezhnev did not seem to understand what Nixon did that necessitated his resignation and express fear that it would undo the "great progress" the US and the USSR made together under detente. The Soviets feared that the loss of a partner in the US might jeopardize US-Soviet relations in the same way that FDR's death in 1945 brought in a president more hostile to the USSR.
The Soviet political culture, with each weak institutional and legal checks on executive power, meant that its leadership had trouble coming to grasps with the idea that the president could be forced to resign over abuses of power in contradiction of the law. Most Soviet officials considered the whole impeachment process not as a matter of constitutional constraint on presidential power, but rather a conspiracy of ultra-rightists who opposed detente, newspapers, and disgruntled democrats to depose Nixon. While others, such as the Soviet deputy procurator general, thought the whole affair was "for show" and that Nixon would survive if he just "showed a little firmness".
Soviet Americanists are however, are at least closer to the mark. The Soviet political scientist Valentin Zorin described Watergate as a function of a power struggle between congress and the president, each of whom represented certain economic "monopoly groupings" power bases inside the US. Watergate was a reaction against the increasing power of the presidency and Nixon's powerbase and attempted to "correct" the balance of power in American politics. The political scientist S. B. Chetverikov described it as a power struggle to contain the power of an autocratic presidency. The Americanists in the USSR broadly perceived Watergate as a reaction against the overly autocratic tendencies of one president, which they understood through their own history with Stalin.
The one way Watergate did change Soviet perception of the US is to give them more respect to the power of congress as well as the constitution to guard against the power of the president. However, to this day, relatively little has being written in the USSR or Russia regarding the Watergate affair.
Source: Soviet Perceptions of the United States By Morton Schwartz