Chancellor Gordon Brown called on those holding the reporter to free him "as a matter of urgency" as they were "not serving their cause by detaining him in this unfair and unjust way".
In a statement, the Foreign Office said: "We condemn the release of videos like this, which can only add to the distress of Alan Johnston's family and friends."
A statement released by Mr Johnston's family in Scotland said relatives were "very pleased to see Alan and to hear him say that he is not being ill-treated - although it is clearly distressing for us to see him in these circumstances".
Mr Johnston, 45, was seized nearly 12 weeks ago in Gaza City on his way home.
It is unclear when the video, which has appeared on the al-Ekhlaas website, was recorded.
Nor was it clear under what conditions the Scottish reporter was speaking.
On the video Mr Johnston says: "First of all, my captors have treated me very well... They have fed me well, there has been no violence towards me at all and I'm in good health."
Mr Johnston, seated and wearing a red sweatshirt, calls for an end to Western sanctions that have been imposed on the Palestinian government.
ARMY OF ISLAM FACTS
Small, Islamist armed group operating in Gaza
Splinter group of the Popular Resistance Committees
Seeks liberation of Palestine and an Islamic state
Influenced by, but not affiliated with, al-Qaeda
Led by Mumtaz Dugmush, also known as Abu Muhammad, a member of a powerful clan
One of three groups allegedly holding captured Israeli soldier Cpl Gilad Shalit
Mr Johnston talks of the "huge suffering" of the Palestinian people, saying: "Everyday there are Palestinians arrested, imprisoned for no reason. People are killed on a daily basis. The economic suffering is terrible, especially here in Gaza."
He also says the British government is working to occupy Muslim lands against the will of the people there.
Mr Johnston refers to the "failed invasion of Iraq by America and Britain" and the "terrible" situation in Afghanistan.
Towards the end of his portion of the video, Mr Johnston begins a message to his family, saying, "to my family, to my family..." but the audio is then cut off.
A message then appears on screen in English saying: "BBC refused to take this message to his family". The BBC's Bridget Kendall says it appears that the kidnappers added this to the video.
In response the BBC said it had kept the family informed throughout and the Johnston family statement said relatives had been "fully informed and involved by the BBC since the day Alan was abducted".
The BBC statement read: "This is a highly distressing time for them and for his friends and his colleagues. We repeat our call for his immediate release."
On the tape, the Army of Islam demand the release of Abu Qatada, a Palestinian-born Islamic cleric who is suspected of close links to al-Qaeda and is currently held by the UK government as a threat to national security.
Senior Palestinian negotiator Saeb Erekat again called for the unconditional and immediate release of Mr Johnston and said that all Palestinians stood behind him.
Mr Johnston was the only Western reporter permanently based in Gaza and his abduction has triggered appeals for his release from lawmakers and rights groups around the world.
More than 130,000 people have now signed an online petition calling for his release.