Constitutional relations secretary Mike Russell has claimed that the lack of time and information provided for EU citizens to complete the paperwork needed robbed many of the vote.
It comes after widespread reports that hundreds of EU nationals were turned away from UK polling stations because of confusion over additional requirements to register for European elections.
Mr Russell has written to Cabinet Office David Lidington and the Electoral Commission calling for action.
"I believe it is imperative that an inquiry is conducted into these issues as a matter of urgency," he said.
"Any seat that is secured by only a small number of votes could be impacted by this outrageous deprivation of democratic rights."
Mr Russell says there was always a risk that the UK would have to take part in the European Parliament Election if the UK was still in the EU on 31 March 2019.
"Despite the clear and obvious signs that this was highly likely to be the case, no preparations were made until the last possible moment," he adds.
To be able to vote, EU citizens had to complete the EC6 or UC1 forms to declare they would not cast a second ballot in their home country, and submit them to their local council by 7 May.
The Electoral Commission said the “very short notice” that the UK would participate in EU elections meant it had not been able to properly publicise the extra requirement.
Mr Russell says is aware of "first-hand reports" of EU citizens being unable to vote even though they had completed and returned the forms on time.
"I would ask you to investigate the scale of this problem which may have arisen due to the lack to time in the run up to the election," his letter adds.
"It is a disgrace that in addition to the uncertainties that EU citizens have had to endure, over the last almost 3 years since the EU referendum, some have been denied their right to vote in the European Parliament Election.
"We want EU citizens in Scotland to feel settled and secure, and to continue to feel welcome and valued in Scotland. This debacle will do nothing to ease their concerns."