Yes and no.
Considered as an abstraction, public opinion can be ignored by ruthless men in power, it's true. See the US today.
But insofar as public opinion influences parliaments, kings, prime ministers, presidents, those determining national policy, public opinion may be everything. For example we know that public opinion averse to the Romanovs, at least among Britain's laboring classses, led to the decision on the part of George V and Parliament to deny the Romanovs asylum in Great Britain.
In the case of public opinion in other countries during the Russian Civil War, it helped , at least for a while, to lead to the decision on the part of the US, Great Britain, France, and Japan, to intervene against the Bolsheviks in Northern Russia and in the Vladivostok area and eastern Siberia. This was obviously a serious political and military problem for the Bolsheviks and one they would not have wanted to make worse by arousing hostile public opinion in those countries.
But more importantly, public opinion, considered more generally as popular support or popular opposition, mattered tremendously to the Bolsheviks in the spring and summer of 1918, as it did to the Whites. If you're recruiting fighting men among the local populace and also needing to be fed by them, as were the Bolsheviks, well,it doesn't pay to offend them or unnecessarily piss them off , or ignore their opinions, so to speak. Exactly how the murder of the IF would play to these people certainly varied, but it's hard to see how it would have benefited them. Opposition to Nicholas' reign most certainly doesn't equate to approval of the murder of his family.
Obviously public opinion is not monolithic or static, and varies from place to place. And in the RUssian instance we're talking about , it certainly wasn't widely pro-Romanov inthe Urals. But neither was it (or wouldn't have been) approving of the slaughter of the entire IF, females and children included, especially among the western democracies , nor even within Russia generally, and again, even less so among Orthodox Russians, including the peasantry, whose opinion, favorable or otherwise, it would not be wise to ignore.