Lord Woolman refused the challenge from The Congregation of the Poor Sisters of Nazareth and the Daughters of Charity of St Vincent De Paul to the Scottish Government's choice of Susan O'Brien to lead the probe.
The charities had raised fears of "apparent bias" relating to her involvement in a previous case brought against the Poor Sisters of Nazareth.
In 2007, two former residents of Nazareth House children's home in Glasgow who alleged they were abused in the 1960s and 70s sought damages from the religious organisation but their claims were held to be "time barred", or made too late.
Ms O'Brien, who is due to take up her post today, represented the two in their subsequent unsuccessful appeal to the House of Lords.
In a hearing at the Court of Session last week Alastair Duncan QC, representing the charities, said: ''The particular concern that my clients have is that Ms O'Brien had acted for individuals alleging abuse against them, that she had supported the allegations that were made by appearing as counsel for those individuals and that she is now being asked to adjudicate on the very same issues.''
Mr Duncan said the allegations that arose in the House of Lords case were ''almost certain'' to be heard again at the public inquiry.
He also told the court that Ms O'Brien had acted in a number of other cases in connection with allegations of historic abuse.
Rejecting the petition for judicial review, Lord Woolman said he did not accept that Ms O'Brien had "supported" her clients' claims in the way suggested by Mr Duncan.
He said: "She only acted for the claimants at a very late stage to argue a point of law. She simply advanced her clients' case to the court.
"Ms O'Brien's narrative of her career discloses that she had a wide-ranging practice covering many areas of the law. She represented pursuers and defenders.
"In particular, she acted both for and against the government. She could not be identified with only one side."
He concluded that Ms O'Brien did not have a close association with an interested party to the inquiry and that the charities had failed to establish apparent bias.
Education Secretary Angela Constance announced Ms O'Brien's appointment last month with the inquiry due to get under way in October.
It will cover allegations of abuse of children in formal institutional care including faith-based organisations, children's homes and secure care as well as those in foster care, long-term hospital care and boarding schools.
The inquiry will have the power to compel witnesses to attend and give evidence.
The Education Secretary previously pledged that where crimes are uncovered the ''full force of the law'' would be used to bring those responsible to justice.
Survivors of abuse were consulted on who should lead the inquiry after two people appointed to chair a similar UK inquiry both stood down.
A Scottish Government spokesman said: "We welcome Lord Woolman's decision, which means that Ms O'Brien can continue the work she began this morning as the chair of the inquiry."