Mervan Celik - who was substituted after just 45 minutes of his first start in the 2-0 defeat - was the club's only capture of the January window despite the sale of top goalscorer Nikica Jelavic to Everton on transfer deadline day.
With manager Ally McCoist unable to secure a replacement before the deadline, Davis knows free agents are the only option now but would like to see the manager given the green light to make some new additions.
The midfielder said: "The window is closed and it's only frees that are going to be available but it would be nice to see a bit of investment in the team and try to add to what we've already got.
"The problem is that we've got a lot of key players out as well and that's something that's gone against us this season."
Rangers' financial plight has been well-documented as they await the verdict of a tax tribunal that could leave them with a bill of £49million if they lose a dispute with HMRC.
Davis added: "That's been hanging over us for a long period of time now.
"But it's disappointing to see players like Jelavic - a very important player - leave without really getting anybody in.
"That was a bit of a blow for the players."
As well as the Scottish Cup, Rangers have also exited the Champions League, the Europa League and the Scottish Communities League Cup.
Their only hope of silverware now is the Clydesdale Bank Premier League title but - despite once boasting a 15-point lead - the Scottish champions now trail Celtic by a point having played a game more.
Asked if Rangers' season is in danger of crumbling altogether, Davis said: "It's up to everybody in the dressing room to make sure that doesn't happen.
"I think the league is still there for us, without doubt.
"Sometimes it's hard to reflect on it straight after the game when everyone is a little bit down and it's hard to see the positives to take out of the situation.
"But we will get back to work, roll our sleeves up and get ready to go again.
"We've had a lot of success over the last number of years but now is the time, when things are going against you, for everybody to be pushing in the same direction.
"We need to give it our all and show the desire and what it means to pull on that shirt."
Reflecting on the defeat to United, the Northern Irishman added: "It's quite simple - it just wasn't good enough.
"Overall, we got what we deserved and that was nothing."
First half goals from Gavin Gunning and Johnny Russell secured a fifth-round win for United.
Russell said: "Rangers are always going to be difficult and they worked hard but it just didn't come off for them and we took our chances and came away with the win."
The striker was recently linked with a move to Celtic but says he is happy to remain at Tannadice for now.
He added: "It's good to hear that clubs are interested in you, especially massive clubs like Celtic.
"But I think the best place for me just now is Dundee United. I've still got a lot to learn and hopefully I can do it here."
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