Much like the sadhu that is allowed to skin the tiger if it died naturally, is it written anywhere in scripture that eating flesh from an animal that died naturally is allowed?
Manu Smriti 5.48. Meat can never be obtained without injury to living creatures, and injury to sentient beings is detrimental to (the attainment of) heavenly bliss; let him therefore shun (the use of) meat.
From the above, it is clear that the reason why meat eating is not recommended is because it involves Ahimsa (injury to other beings).
But, when the animal has naturally died, there is no Ahimsa involved and the above prohibition or reasoning is not applicable.
Is eating meat allowed if the animal died naturally?
From the above discussion it should not be prohibited. But there can be other concerns regarding eating such meat like hygiene or others. Because a being dies naturally out of old age or diseases.
However, note that the verse given above does not even take into consideration the option that "meat can be obtained WITHOUT injuring the animal". So, eating meat of an animal which has naturally died may not be even an option as per the scriptures.
Medhâtithi's commentary on the 26th verse of 5th chapter of Manusmriti :
The first half of the verse cuts off the preceding section; and what is implied by this cutting off of the section is that the section that has gone before pertains to the twice-born castes only, not to Śūdras, while what follows applies to Śūdras also. It is for this reason that several methods of eating meat shall be described, and the reward resulting from the giving up of meat-eating shall accrue to the Śūdra also. If this were not so, then, in the matter of eating meat also, the Śūdra would be free to do what he likes; just as he is in regard to the eating of garlic and other things that has been forbidden for ‘twice-born persons’ only, in verses 5 ete. etc. above. “If it is as you say, then there is the following difficulty:—In verse 32 below, the Text is going to declare the eatability of the meat left from the worship of the Devas:—viz. ‘One does not become contaminated by sin if he eats meat after having worshipped the Devas and the Pitṛs’;—now the ‘worship of the Devas’ etc. can be done only with such meat which is sacred; and those beasts and birds that have been forbidden for twice-born people (in the next section) are not sacred; hence, the worshipping of Devas etc. with the meat of these beasts and birds being impossible,—and what does not form the ‘remnant of worship’ being unfit to be eaten,—these other beasts and birds also, mentioned in a different context, become forbidden for the twice-born people; and the prohibition of these could be made to apply to the Śūdra also by some such other method (of reasoning). So that there is no point in the dividing of the sections (simply for making the prohibitions of the next section applicable to Śūdras also). And as for the prohibition of garlic and such things (that have been forbidden specially for twice-born persons), it is not applicable to Śūdras at all.” There is this useful purpose served by the dividing of the two sections, that the prohibition of garlic and other things ceases to be applicable to the Śūdra. As regards meat also, in as much as the Householder only is entitled to do the worshipping of Devas, it is a matter purely optional for such Śūdras as are not ‘householders’. “As a matter of fact, Śūdras also are entitled to the performance of sacrifices with cooked food; the eating of food has also been prescribed for Householders; but no ‘sacrifices with cooked food’ are ever offered with garlic and such other things. So that these things may be eaten, or not, by Śūdras, entirely according to their option.—‘Why’? What would be the harm? In that case the mention of ‘twiceborn persons’ (in connection with the forbidding of garlic, etc.) would have no point at all.’ This has been already answered by the explanation that one who is not a Householder, or who is travelling away from home, may do what he likes. Nor is it necessary that the Householder shall not eat what has not been offered in oblations; the meaning of the declaration ‘one shall live on remnants’ being that ‘he shall not eat until he has made the offering to the Visvadevas.’ Now, that substance alone is ‘sacred’, and can be offered as oblation, which has been prescribed as to be offered at, and thus helping the fulfilment of, a sacrifice. Some people fetch food from somewhere, at the time of eating, and eat it in their own house; and in this case even though the food may not be the ‘remnant of a sacrifice’, it would not be forbidden. As regards meat however, we have the restriction directly imposed, that ‘it shall never be eaten unless it has been offered to the Devas.’ If this is applicable to all the four castes, then there is no point in what is going to be said (under 57th verse) in connection with purifications.” The use of that we shall explain at that place. In view of the mention of twice-born people in the foregoing section, it follows that day’s meat and such things also are fit to be eaten by Śūdras.” Under the following discourses, we shall show that there are indications to the effect that the village-pig, the ass, the camel, and other animals mentioned in the three verses (157 etc.) are ‘unfit to be eaten’ for the Śūdra also.
He may eat meat that has been consecrated; also at the wish of Brāhmaṇas; and when invited according to law; and when his life is in danger.[Manusmriti:5:27]
The eating of meat for sacrifices’—this is declared to be the divine law; but behaviour contrary to this is described as ‘demonical practice'.[Manusmriti:5:31]
Medhâtithi's commentary on this verse :
'The eating of meat'—in the form of offerings and oblations —'for sacrifices.’ 'This is the divine law’—this is what has been ordained by the Devas. ‘Behaviour contrary to this,’—i.e. eating meat for the fattening of the body—is ‘the demonical practice’; it is only demons that eat meat in this fashion. This is said in deprecation of the practice.
Having bought it, or having obtained it himself, or having it presented by others,—if one eats meat after having worshipped the Devas and the Pitṛs, he does not incur sin.[Manusmriti:5:32]
Medhâtithi's commentary on this verse :
The law here laid down refers to the meat of deer and birds. The meaning is that there is no sin incurred in eating the meat of the Ruru, the Pṛṣat and other kinds of deer, or the partridge and other birds, if it is done after having worshipped the Devas and the Pitṛs. In the case of the offering to the Viśvedevas, when there is no preparation for it in the house, one may eat rice and other things, even without making the offering; but not so meat; it is with a view to emphasise this that the text repeats—‘having worshipped the Devas and the Pitṛs etc.’ If mere sanction to eat after worship were meant, this has already been accorded before. What is meant by the ‘worshipping of the Devas’ here is the offering of the meat on a clean spot with the words ‘this is for the Devas’; or that ‘the worship of the Devas’ should be done in such terms as—‘this to Agni, to Vāyu, to Surya, to Jātavedas.’ That this must be the meaning of the ‘worship’ is proved by the fact that ‘offering of oblations into the fire’ (which could be the other meaning ‘worshipping the Devas’) is not possible for persons other than Agnihotrins; nor can there be any offering made to the Devas without oblations having been poured into the fire; specially as it has been already shown that the two are distinct actions and involve distinct methods of procedure. This matter may rest here for the present. Others have explained the ‘worship of the Pitṛs’ to mean Śrāddha; and in Śrāddhas we do find worshipping being done. It is the Pitṛs again that are spoken of as the ‘deities’ of the Śrāddha. Hence it is that in connection with the Pitṛs, all writers on Smṛti have prescribed the Śrāddha only, and no other act. “How can the buying of meat be permissible? The meat obtained from the market becomes ‘butcher’s meat’ (which has been forbidden); and as for the meat of animals dying of themselves, and not killed by the butcher, this is ‘unfit for eating’, on the ground of its causing disease.” Our answer to the above is that one can always ‘buy’ the meat brought by fowlers and bird-catchers; and these are known as ‘butchers’; and they wander about from house to house, carrying meat for sale, when it is possible to buy it; and it does not become ‘butcher’s meat.’ ‘Having obtained it himself,’—the Brāhmaṇa by begging it’ and the Kṣatriya by hunting.