Independent barrister Katharine Newton concluded in her final report - published on Wednesday - that Sampson was not racist, but that he twice made "ill-judged attempts at humour" towards the England pair.
Football Association chief executive Martin Glenn has "sincerely apologised" to Aluko and Spence over Sampson's remarks, calling them "not acceptable".
The report found no evidence, however, to support allegations that Aluko was subjected to "a course of bullying and discriminatory conduct" by the then national team boss.
Sampson was dramatically sacked last month after FA chiefs were alerted to what it termed an "inappropriate" relationship he had with a player in his previous job in 2013.
That followed weeks of speculation about his position after it emerged he had already been the subject of two FA investigations into allegations of discrimination by Chelsea striker Aluko.
Aluko claimed Sampson had told her to make sure her Nigerian relatives did not bring the Ebola virus to the friendly against Germany at Wembley in November 2014.
Sampson denied that claim, along with another allegation that he asked a mixed-race player - Spence - if she had been arrested before, and then jokingly suggested she had been arrested four times.
Glenn said in a statement: "On behalf of the Football Association I would like to sincerely apologise to Eniola Aluko and Drew Spence.
"Based on new evidence submitted to independent barrister Katharine Newton, she has now found that they were both subject to discriminatory remarks made by an FA employee. This is not acceptable."
Aluko was appearing before the Digital, Culture, Media and Sport committee in Westminster on Wednesday afternoon.