After the assault on the Citadel, I do my best not to shoot Ashley. However, the options for saving her seem to be grayed out.
Is there any way to save her? What actions do I have to undertake to make these options available?
There's actually a host of factors that influence the outcome of this interaction. What it comes down to is whether the Survivor trusts you. Whether you've built up enough trust with them will determine whether they will believe you outright when you confront them, require a hard, or possibly even max level Charm/Intimidation check to persuade (this appears to be what happened to you, but sadly, you don't have enough Paragon/Renegade points to make it through), or if you've been truly neglectful and/or hostile, they may be unable to be swayed.
These factors include:
Edit 2:
Finally going through this part of the game myself (Romancing Ashley in 1) I had a different set of dialogue choices that were Paragon/Renegade. Going with the Paragon interrupt (have your squad stand down) and then choosing the Paragon dialogue option, I saved Ash's life. I can only assume if I did not do the interrupt and/or chose the Renegade option, she would not have survived.
Ashley is unaware that Udina and Cerberus are working together to lead the Councilors into a trap so Udina can take control of the Council. This puts Shepard and Ashley in conflict; Shepard must then convince Ashley into realizing the truth about what's going on and turning on Udina instead. If Shepard cannot persuade Ashley to back off, then Shepard can kill Ashley. If not, a squad member will shoot her.
Edit:
In this walkthrough, it shows that it seems the initial dialogue choice and whether or not you choose the Paragon interrupt determines if Ashley/Kaiden lives or dies during this encounter. This walk through also did not Romance Ashley in ME1, but Liara, which explains the different dialogue options than mine above.