(RTD vs Moffat alert...)
Although I've enjoyed (and been dismayed by) both writing styles, I've long thought that Russell's character writing was more realistic than the Moff's in Doctor Who. Structurally and in essence. I don't think sexism is really the problem but a side effect - one that is highlighted if you focus absolutely on female-only interaction.
I think critics of RTD would dub it the 'soap' element but I would call it social realism. Rose, Jackie, Mickey, Martha (to an extent), Donna, [her mum], Wilf: all had frequently been given dialogue which demonstrated their ties to the everyday, which neatly contrasted with the Doctor's life.
If a companion (the 'anchor' for the audience) is given an improbable fairytale-like background (the Girl Who Waited; the Impossible Girl; the Woman who Kills the Doctor; the Last Centurion) and has his / her timeline routinely erased and rewritten to the point of nausea from the smell of Tipex, the Doctor suddenly becomes the more accessible character...which isn't really how it should be.
How does this relate to sexism? The Doctor is written as a prominent figure in each of the Moffat companions' existence (quite literally from conception in River's case). There's no quarrel there: in fact, it is rather interesting to explore how much of an effect the Doctor can have on individuals. Regardless, those characters (with the exception of Clara) are psychologically predisposed to talk about a dominant presence in their lives and that dominant presence happens to be a 'man'. The Doctor is a legend, a warrior, a problem, the pointiest of plot points...and he's a man. If you make the show as well as its central characters gravitate around this figure continuously, then the side effect is going to be an overload of the word 'he'.
If this were to be considered a problem, there are several solutions:
1. More two-parters or an extended running time for episodes so characters can be given more Doctor-free dialogue to provide depth.
2. More women writers: chances are the female characters would be written more realistically, though I realise this presumption is probably sexist in itself. The irony!
3. Have the companion(s) or secondary characters take centre stage more frequently, relegating the Doctor to the sidelines. Some would say this goes against the premise of the show but it has been done many times, at least in part.