I understand from Dudziak's "Cold War Civil Rights" that Soviet criticism of the US's civil rights record influenced the passing the 1964 Civil Rights Act. However, I was wondering if there were other ways that the Cold War contributed to the bill's passing. For instance, did the American public feel ashamed of its racial crises in the midst of the Cold War's ideological war, leading to public support for the bill? Did the US government, seeing the movement as communist, seek to appease it before things escalated?
As I have pointed out in the comments, civil rights are quite liberal/capitalist concept, so emancipation had to happen eventually, even without any pressure exerted by the USSR.
However, the USSR did support some leading figures in the civil rights movement: e.g., Angela Davies was twice a vice-presidential candidate from the Communist Party, and actively mingled with Soviet and other East European leaders. In this sense, passing the Civil Rights act was likely making the Communists movements less relevant, and served to diminish their influence.
There were a number of things contributing to the push for civil rights up to 1964.
For a start, the Americans themselves had fought a civil war in the 1860s largely over the status of black slave labour.
The practice of slavery had already been outlawed in Europe since time immemorial, and America was a European settler colony. So the USA was already on a long trajectory towards establishing the same civil rights as prevailed in Europe.
By the 1930s, the USSR was really at the forefront of developing what we nowadays think of as ideal "liberal rights" enjoyed by all citizens. It was a political union that drew together a large number of ethnic communities (at a time when racism in many nations was otherwise at a high), and as an aside it also promoted the rights and capabilities of women as workers too.
This also had an influence on how for example Britain mobilised women into industry at the outbreak of WW2.
For the American civil rights movement however, there were several consequences of WW2 itself.
Firstly, black soldiers were mobilised abroad in large numbers, and this broadened their horizons and often exposed them to European populations (like the British) who did not generally have the prejudicial attitudes of the American south (which was still frankly reeling economically from the civil war, and plagued by everyday racism).
Secondly, as veterans having fought for the American state, a new generation not only were equipped with military skills, but also naturally developed a sense of confidence and moral righteousness in demanding equal treatment.
Thirdly, at the conclusion of WW2, racist ideologies amongst ruling classes had been dealt a terrible blow, not only by the defeat of Hitler as one of its main representatives, but by the horrors perpetrated by his regime in the meantime. It became unpopular and clearly dangerous to the ruling classes themselves to keep promoting the lies of racial ideologies which both provoked uncontrollable conflicts and only weakened those who adhered most strongly to those lies.
In the post-WW2 era when the USA was really becoming concerned about the influence of communism and the USSR, together with the increased endogenous strength of the civil rights movement for the reasons just stated, this was why progress towards better civil rights became irresistible, and the resistance against no longer tenable.