Which complexes are stable: ammine complexes or carbonyl complexes?
That quotation from Wikipedia is bad and should be reworded (I can’t think of good ways to edit it, though). Ligands in a spectrochemical series are not ordered by bonding strength or anything, they are ordered solely by the field split they induce.

Both $\ce{CO}$ and $\ce{NH3}$ are great at forming complexes — as is iodide. The main difference is that they won’t form similarly strong complexes with the same cores. Just to give a few examples:


Carbon monoxide is neutral and a &#x3c0;-acid: It has an empty orbital with &#x3c0; symmetry which can take part in metal-to-ligand back-bonding. As such, it stabilises low oxidation states of metals. The neutral carbonyl complexes such as $\ce{[Fe(CO)5]}$ are well known, but also complexes such as $\ce{[Fe(CO)4]^2-}$ (yes, that’s iron($\mathrm{-II}$)!) are stable.
Ammonia is also neutral and a very weak &#x3c0;-base. However, it is a much stronger Brønsted base than carbonyl is. Therefore, it is much better at stabilising medium to high oxidation states such as in $\ce{[Cu(NH3)4(H2O)2]^2+}$.
The halides, finally, are anionic and good &#x3c0; bases and thus able to stabilise high oxidation states. They are classed as weak-field ligands because they generally generate a very low field split. An example of a halide complex would be the bright yellow $\ce{[FeCl6]^3-}$.


You cannot take ligand $\ce{X}$ and call it strong or weak and thus expect it to make stable or unstable complexes. You cannot even take the same ligand and expect it to make inert or labile complexes all the time. One ligand-metal combination may be stable, another may be very unstable — $\ce{[Fe(CO)6]^2+}$ can only be synthesised under a $\ce{CO}$ atmosphere in $\ce{SbF5}$ from $\ce{FeF2}$ where the antimony pentafluoride abstracts any free fluoride ions from the solution leaving a ‘naked’ iron ion which will grab the only ligands it can reach. Iron pentacarbonyl is stable. The same goes for ammonia: Some will readily accept it as a ligand, others won’t. There is no black and no white, only shades of grey.

(Although since coordination compounds are generally colourful let me rephrase that: There is no black and no white, only a wheel of colours.)