The Deputy First Minister faced questions in Holyrood as families said they already felt failed by the review, which has yet to hold a single session.
Its lead counsel and three junior counsels quit on Thursday, while Lady Poole, its chair, resigned the following day for "personal reasons".
Mr Swinney said he was made aware of Lady Poole’s decision during a telephone call on Friday morning.
He told MSPs: "In the course of that call she indicated to me that four members of counsel had resigned the previous day from the inquiry.
"That was news to me, as were the circumstances that led to Lady Poole's resignation when I heard that on Friday morning."
Alan Wightman, whose 88-year-old mother Helen died in 2020 during a Covid outbreak at her care home in Fife, said he was "disappointed" by the resignations.
Asked if he felt families had been failed, he told the BBC: "I would have to say at the moment, yes I do because it has achieved absolutely nothing in the 10 months it has been in place, other than some consultation with the chair of the UK inquiry who knew what she was doing."
Mr Wightman leads the Scottish chapter of Covid-19 Bereaved Families for Justice.
Lawyer Aamer Anwar, who represents the group, said the inquiry looks “like a sinking ship” and families felt “betrayed”.
Speaking in Holyrood, Scottish Labour deputy leader Jackie Baillie questioned why MSPs were not told of the four legal team resignations when news of Lady Poole's decision was released by the Scottish Government.
She said: "Some more cynical than I might say that there's a pattern of secrecy here with the Government and I hope that this doesn't spill over into the inquiry itself."
Mr Swinney insisted the inquiry is legally independent of the Scottish Government.
He said: "At no stage have I tried to conceal information. I've simply respected the legal framework under which I must operate."
Asked if the inquiry might be delayed, he said the Scottish Government wanted it to be delivered "at speed".
The Deputy First Minister said Lady Poole would continue as chair during a notice period of up to three months.
Ms Baillie asked when the inquiry will start, when it will hear from families, when it will report and what its revised costs are. Mr Swinney said answering those questions would "break the law" by interfering in its operation.
Scottish Conservative MSP Murdo Fraser meanwhile pressed on whether Lady Poole's resignation was prompted by the four counsel members quitting, and asked for an "assurance that none of these resignations have come about because of any political interference in the direction or operation of the inquiry".
The Deputy First Minister responded: "Lady Poole indicated to me that for personal reasons she wished to step down from the inquiry.
"Those were Lady Poole's words to me and that is what she has indicated to me, and I don't think it's incumbent on me to explore or examine the rationale for Lady Poole's statements to me.
"The final point is frankly one to which I take the greatest of exception because I have judiciously followed the contents of the Inquiries Act, and particularly section 17, which guarantees the independence of the inquiry.
"And just for the record, there has been absolutely no political interference in the inquiry."
Insiders reportedly suggested that relations between the lawyers and Lady Poole had broken down.
The inquiry will look at the Scottish Government's actions during the pandemic and issues such as the transfer of hospital patients into care homes.
The UK equivalent opened in London on Tuesday.
A spokesman for the Scottish Covid-19 inquiry said: “We can confirm that four members of the counsel team have left the inquiry.
“We are immensely grateful for their contribution to the inquiry’s progress and wish them every success.”